Distribución normal

Probability distribution

En teoría de probabilidad y estadística , una distribución normal o distribución gaussiana es un tipo de distribución de probabilidad continua para una variable aleatoria de valor real . La forma general de su función de densidad de probabilidad es El parámetro es la media o esperanza de la distribución (y también su mediana y moda ), mientras que el parámetro es la varianza . La desviación estándar de la distribución es (sigma). Se dice que una variable aleatoria con una distribución gaussiana tiene una distribución normal y se denomina desviación normal . $${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}$$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle \sigma }$

Las distribuciones normales son importantes en estadística y se utilizan a menudo en las ciencias naturales y sociales para representar variables aleatorias de valor real cuyas distribuciones no se conocen. ^{[2] [3]} Su importancia se debe en parte al teorema del límite central . Este afirma que, en determinadas condiciones, el promedio de muchas muestras (observaciones) de una variable aleatoria con media y varianza finitas es en sí mismo una variable aleatoria, cuya distribución converge a una distribución normal a medida que aumenta el número de muestras. Por lo tanto, las cantidades físicas que se espera que sean la suma de muchos procesos independientes, como los errores de medición , a menudo tienen distribuciones que son casi normales. ^[4]

Además, las distribuciones gaussianas tienen algunas propiedades únicas que son valiosas en los estudios analíticos. Por ejemplo, cualquier combinación lineal de un conjunto fijo de desviaciones normales independientes es una desviación normal. Muchos resultados y métodos, como la propagación de la incertidumbre y el ajuste de parámetros por mínimos cuadrados ^[5] , se pueden derivar analíticamente en forma explícita cuando las variables relevantes se distribuyen normalmente.

A veces, a una distribución normal se la denomina informalmente curva de campana . ^[6] Sin embargo, muchas otras distribuciones tienen forma de campana (como la distribución de Cauchy , la t de Student y las distribuciones logísticas ). Para conocer otros nombres, consulte Nomenclatura .

La distribución de probabilidad univariante se generaliza para los vectores en la distribución normal multivariante y para las matrices en la distribución normal matricial .

Definiciones

Distribución normal estándar

El caso más simple de una distribución normal se conoce como distribución normal estándar o distribución normal unitaria . Este es un caso especial cuando y , y se describe mediante esta función de densidad de probabilidad (o densidad): La variable tiene una media de 0 y una varianza y desviación estándar de 1. La densidad tiene su pico en y puntos de inflexión en y . ${\textstyle \mu =0}$ ${\textstyle \sigma ^{2}=1}$ $${\displaystyle \varphi (z)={\frac {e^{\frac {-z^{2}}{2}}}{\sqrt {2\pi }}}\,.}$$ ${\textstyle z}$ ${\textstyle \varphi (z)}$ ${\textstyle {\frac {1}{\sqrt {2\pi }}}}$ ${\textstyle z=0}$ ${\textstyle z=+1}$ ${\textstyle z=-1}$

Aunque la densidad anterior se conoce más comúnmente como la normal estándar, algunos autores han utilizado ese término para describir otras versiones de la distribución normal. Carl Friedrich Gauss , por ejemplo, una vez definió la normal estándar como que tiene una varianza de ⁠ ⁠ , y Stephen Stigler ^[7] una vez definió la normal estándar como que tiene una forma funcional simple y una varianza de $${\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},}$$ ${\displaystyle {\frac {1}{2}}}$ $${\displaystyle \varphi (z)=e^{-\pi z^{2}},}$$ ${\textstyle \sigma ^{2}=1/(2\pi ).}$

Distribución normal general

Toda distribución normal es una versión de la distribución normal estándar, cuyo dominio ha sido ampliado por un factor (la desviación estándar) y luego traducido por (el valor medio): ${\textstyle \sigma }$ ${\textstyle \mu }$

$${\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)\,.}$$

La densidad de probabilidad debe escalarse de modo que la integral siga siendo 1. ${\textstyle 1/\sigma }$

Si es una desviación normal estándar , entonces tendrá una distribución normal con valor esperado y desviación estándar . Esto es equivalente a decir que la distribución normal estándar se puede escalar/estirar por un factor de y desplazar por para producir una distribución normal diferente, llamada . Por el contrario, si es una desviación normal con parámetros y , entonces esta distribución se puede volver a escalar y desplazar mediante la fórmula para convertirla en la distribución normal estándar. Esta variante también se denomina forma estandarizada de . ${\textstyle Z}$ ${\textstyle X=\sigma Z+\mu }$ ${\textstyle \mu }$ ${\textstyle \sigma }$ ${\textstyle Z}$ ${\textstyle \sigma }$ ${\textstyle \mu }$ ${\textstyle X}$ ${\textstyle X}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle X}$ ${\textstyle Z=(X-\mu )/\sigma }$ ${\textstyle X}$

Notación

La densidad de probabilidad de la distribución gaussiana estándar (distribución normal estándar, con media cero y varianza unitaria) se denota a menudo con la letra griega ( phi ). ^[8] La forma alternativa de la letra griega phi, , también se utiliza con bastante frecuencia. ${\textstyle \phi }$ ${\textstyle \varphi }$

La distribución normal a menudo se denomina o . ^[9] Por lo tanto, cuando una variable aleatoria se distribuye normalmente con media y desviación estándar , se puede escribir ${\textstyle N(\mu ,\sigma ^{2})}$ ${\textstyle {\mathcal {N}}(\mu ,\sigma ^{2})}$ ${\textstyle X}$ ${\textstyle \mu }$ ${\textstyle \sigma }$

$${\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}$$

Parametrizaciones alternativas

Algunos autores recomiendan utilizar la precisión como parámetro que define el ancho de la distribución, en lugar de la desviación estándar o la varianza . La precisión se define normalmente como el recíproco de la varianza, . ^[10] La fórmula para la distribución se convierte entonces en ${\textstyle \tau }$ ${\textstyle \sigma }$ ${\textstyle \sigma ^{2}}$ ${\textstyle 1/\sigma ^{2}}$

$${\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.}$$

Se afirma que esta elección tiene ventajas en los cálculos numéricos cuando está muy cerca de cero y simplifica las fórmulas en algunos contextos, como en la inferencia bayesiana de variables con distribución normal multivariada . ${\textstyle \sigma }$

Alternativamente, el recíproco de la desviación estándar podría definirse como la precisión , en cuyo caso la expresión de la distribución normal se convierte en ${\textstyle \tau '=1/\sigma }$

$${\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.}$$

Según Stigler, esta formulación es ventajosa debido a una fórmula mucho más simple y fácil de recordar, y a fórmulas aproximadas simples para los cuantiles de la distribución.

Las distribuciones normales forman una familia exponencial con parámetros naturales y , y estadísticas naturales x y x ² . Los parámetros de expectativa dual para la distribución normal son η ₁ = μ y η ₂ = μ ² + σ ² . ${\textstyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}}$ ${\textstyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}}$

Función de distribución acumulativa

La función de distribución acumulativa (CDF) de la distribución normal estándar, usualmente denotada con la letra griega mayúscula ( phi ), es la integral ${\textstyle \Phi }$

$${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\,.}$$

Función de error

La función de error relacionada da la probabilidad de una variable aleatoria, con distribución normal de media 0 y varianza 1/2 dentro del rango . Es decir: ${\textstyle \operatorname {erf} (x)}$ ${\textstyle [-x,x]}$

$${\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt\,.}$$

Estas integrales no se pueden expresar en términos de funciones elementales y, a menudo, se dice que son funciones especiales . Sin embargo, se conocen muchas aproximaciones numéricas; consulte más información a continuación.

Las dos funciones están estrechamente relacionadas, a saber:

$${\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]\,.}$$

Para una distribución normal genérica con densidad , media y varianza , la función de distribución acumulativa es ${\textstyle f}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$

$${\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]\,.}$$

El complemento de la función de distribución acumulativa normal estándar, , se suele denominar función Q , especialmente en textos de ingeniería. ^{[11] [12]} Da la probabilidad de que el valor de una variable aleatoria normal estándar supere a : . Otras definiciones de la función -, todas las cuales son transformaciones simples de , también se utilizan ocasionalmente. ^[13] ${\textstyle Q(x)=1-\Phi (x)}$ ${\textstyle X}$ ${\textstyle x}$ ${\textstyle P(X>x)}$ ${\textstyle Q}$ ${\textstyle \Phi }$

La gráfica de la función de distribución acumulativa normal estándar tiene simetría rotacional doble alrededor del punto (0,1/2); es decir, . Su antiderivada (integral indefinida) se puede expresar de la siguiente manera: ${\textstyle \Phi }$ ${\textstyle \Phi (-x)=1-\Phi (x)}$ $${\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.}$$

La función de distribución acumulativa de la distribución normal estándar se puede ampliar mediante la integración por partes en una serie:

$${\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]\,.}$$

donde denota el factorial doble . ${\textstyle !!}$

También se puede derivar una expansión asintótica de la función de distribución acumulativa para valores grandes de x mediante la integración por partes. Para obtener más información, consulte Función de error#Expansión asintótica . ^[14]

Se puede encontrar una aproximación rápida a la función de distribución acumulativa de la distribución normal estándar utilizando una aproximación de la serie de Taylor:

$${\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}\,.}$$

Cálculo recursivo con desarrollo de la serie de Taylor

La naturaleza recursiva de la familia de derivadas se puede utilizar para construir fácilmente una expansión de la serie de Taylor rápidamente convergente utilizando entradas recursivas sobre cualquier punto de valor conocido de la distribución : ${\textstyle e^{ax^{2}}}$ ${\textstyle \Phi (x_{0})}$

$${\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\,,}$$

dónde:

$${\displaystyle {\begin{aligned}\Phi ^{(0)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt\\\Phi ^{(1)}(x_{0})&={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}\\\Phi ^{(n)}(x_{0})&=-\left(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0})\right),&n\geq 2\,.\end{aligned}}}$$

Utilizando la serie de Taylor y el método de Newton para la función inversa

Una aplicación de la expansión de la serie de Taylor anterior es utilizar el método de Newton para invertir el cálculo. Es decir, si tenemos un valor para la función de distribución acumulativa , , pero no conocemos la x necesaria para obtener , podemos utilizar el método de Newton para encontrar x, y utilizar la expansión de la serie de Taylor anterior para minimizar el número de cálculos. El método de Newton es ideal para resolver este problema porque la primera derivada de , que es una integral de la distribución estándar normal, es la distribución estándar normal y está disponible para su uso en la solución del método de Newton. ${\textstyle \Phi (x)}$ ${\textstyle \Phi (x)}$ ${\textstyle \Phi (x)}$

Para resolver, seleccione una solución aproximada conocida, , para el . deseado puede ser un valor de una tabla de distribución, o una estimación inteligente seguida de un cálculo de utilizando cualquier medio deseado para calcular. Utilice este valor de y la expansión de la serie de Taylor anterior para minimizar los cálculos. ${\textstyle x_{0}}$ ${\textstyle \Phi (x)}$ ${\textstyle x_{0}}$ ${\textstyle \Phi (x_{0})}$ ${\textstyle x_{0}}$

Repita el siguiente proceso hasta que la diferencia entre el calculado y el deseado , que llamaremos , esté por debajo de un error aceptablemente pequeño, como 10 ⁻⁵ , 10 ⁻¹⁵ , etc.: ${\textstyle \Phi (x_{n})}$ ${\textstyle \Phi }$ ${\textstyle \Phi ({\text{desired}})}$

$${\displaystyle x_{n+1}=x_{n}-{\frac {\Phi (x_{n},x_{0},\Phi (x_{0}))-\Phi ({\text{desired}})}{\Phi '(x_{n})}}\,,}$$

dónde

${\textstyle \Phi (x,x_{0},\Phi (x_{0}))}$es la solución de una serie de Taylor usando y ${\textstyle \Phi (x)}$ ${\textstyle x_{0}}$ ${\textstyle \Phi (x_{0})}$

$${\displaystyle \Phi '(x_{n})={\frac {1}{\sqrt {2\pi }}}e^{-x_{n}^{2}/2}\,.}$$

Cuando los cálculos repetidos convergen a un error inferior al valor aceptablemente pequeño elegido, x será el valor necesario para obtener a del valor deseado, . ${\textstyle \Phi (x)}$ ${\textstyle \Phi ({\text{desired}})}$

Desviación estándar y cobertura

Para la distribución normal, los valores menores a una desviación estándar de la media representan el 68,27% del conjunto; mientras que dos desviaciones estándar de la media representan el 95,45%; y tres desviaciones estándar representan el 99,73%.

Alrededor del 68% de los valores extraídos de una distribución normal están dentro de una desviación estándar σ de la media; alrededor del 95% de los valores se encuentran dentro de dos desviaciones estándar; y alrededor del 99,7% están dentro de tres desviaciones estándar. ^[6] Este hecho se conoce como la regla 68-95-99,7 (empírica) o la regla de 3 sigma .

Más precisamente, la probabilidad de que una desviación normal se encuentre en el rango entre y está dada por Para 12 dígitos significativos, los valores para son: ${\textstyle \mu -n\sigma }$ ${\textstyle \mu +n\sigma }$ $${\displaystyle F(\mu +n\sigma )-F(\mu -n\sigma )=\Phi (n)-\Phi (-n)=\operatorname {erf} \left({\frac {n}{\sqrt {2}}}\right).}$$ ${\textstyle n=1,2,\ldots ,6}$

Para valores grandes , se puede utilizar la aproximación . ${\textstyle n}$ ${\textstyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}$

Función cuantil

La función cuantil de una distribución es la inversa de la función de distribución acumulativa. La función cuantil de la distribución normal estándar se denomina función probit y se puede expresar en términos de la función de error inversa : Para una variable aleatoria normal con media y varianza , la función cuantil es El cuantil de la distribución normal estándar se denota comúnmente como . Estos valores se utilizan en pruebas de hipótesis , construcción de intervalos de confianza y gráficos Q-Q . Una variable aleatoria normal excederá con probabilidad , y quedará fuera del intervalo con probabilidad . En particular, el cuantil es 1,96 ; por lo tanto, una variable aleatoria normal quedará fuera del intervalo solo en el 5% de los casos. $${\displaystyle \Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}$$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ $${\displaystyle F^{-1}(p)=\mu +\sigma \Phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}$$ ${\textstyle \Phi ^{-1}(p)}$ ${\textstyle z_{p}}$ ${\textstyle X}$ ${\textstyle \mu +z_{p}\sigma }$ ${\textstyle 1-p}$ ${\textstyle \mu \pm z_{p}\sigma }$ ${\textstyle 2(1-p)}$ ${\textstyle z_{0.975}}$ ${\textstyle \mu \pm 1.96\sigma }$

La siguiente tabla muestra el cuartil que se encontrará en el rango con una probabilidad especificada . Estos valores son útiles para determinar el intervalo de tolerancia para promedios de muestra y otros estimadores estadísticos con distribuciones normales (o asintóticamente normales). ^[15] La siguiente tabla muestra , no como se definió anteriormente. ${\textstyle z_{p}}$ ${\textstyle X}$ ${\textstyle \mu \pm z_{p}\sigma }$ ${\textstyle p}$ ${\textstyle {\sqrt {2}}\operatorname {erf} ^{-1}(p)=\Phi ^{-1}\left({\frac {p+1}{2}}\right)}$ ${\textstyle \Phi ^{-1}(p)}$

Para números pequeños , la función cuantil tiene la expansión asintótica útil ^{[ cita requerida ]} ${\textstyle p}$ ${\textstyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}$

Propiedades

La distribución normal es la única distribución cuyos cumulantes más allá de los dos primeros (es decir, distintos de la media y la varianza ) son cero. También es la distribución continua con la entropía máxima para una media y varianza especificadas. ^{[16] [17]} Geary ha demostrado, suponiendo que la media y la varianza son finitas, que la distribución normal es la única distribución donde la media y la varianza calculadas a partir de un conjunto de valores extraídos independientes son independientes entre sí. ^{[18] [19]}

La distribución normal es una subclase de las distribuciones elípticas . La distribución normal es simétrica respecto de su media y no es cero en toda la línea real. Como tal, puede no ser un modelo adecuado para variables que son inherentemente positivas o fuertemente sesgadas, como el peso de una persona o el precio de una acción . Dichas variables pueden describirse mejor mediante otras distribuciones, como la distribución log-normal o la distribución de Pareto .

El valor de la distribución normal es prácticamente cero cuando el valor se encuentra a más de unas pocas desviaciones estándar de la media (por ejemplo, una dispersión de tres desviaciones estándar cubre toda la distribución total, excepto el 0,27%). Por lo tanto, puede no ser un modelo apropiado cuando se espera una fracción significativa de valores atípicos (valores que se encuentran a muchas desviaciones estándar de la media) y los mínimos cuadrados y otros métodos de inferencia estadística que son óptimos para las variables distribuidas normalmente suelen volverse muy poco confiables cuando se aplican a esos datos. En esos casos, se debe suponer una distribución de colas más pesadas y aplicar los métodos de inferencia estadística robustos adecuados . ${\textstyle x}$

La distribución gaussiana pertenece a la familia de distribuciones estables que son las que atraen las sumas de distribuciones independientes idénticamente distribuidas, independientemente de que la media o la varianza sean finitas o no. A excepción de la gaussiana, que es un caso límite, todas las distribuciones estables tienen colas pesadas y varianza infinita. Es una de las pocas distribuciones que son estables y que tienen funciones de densidad de probabilidad que se pueden expresar analíticamente, las otras son la distribución de Cauchy y la distribución de Lévy .

Simetrías y derivadas

La distribución normal con densidad (media y varianza ) tiene las siguientes propiedades: ${\textstyle f(x)}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}>0}$

• Es simétrica alrededor del punto que es al mismo tiempo la moda , la mediana y la media de la distribución. ^[20] ${\textstyle x=\mu ,}$
• Es unimodal : su primera derivada es positiva para negativa para y cero sólo en ${\textstyle x<\mu ,}$ ${\textstyle x>\mu ,}$ ${\textstyle x=\mu .}$
• El área delimitada por la curva y el eje es la unidad (es decir, igual a uno). ${\textstyle x}$
• Su primera derivada es ${\textstyle f'(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}$
• Su segunda derivada es ${\textstyle f''(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).}$
• Su densidad tiene dos puntos de inflexión (donde la segunda derivada de es cero y cambia de signo), ubicados a una desviación estándar de la media, es decir en y ^[20] ${\textstyle f}$ ${\textstyle x=\mu -\sigma }$ ${\textstyle x=\mu +\sigma .}$
• Su densidad es logarítmica-cóncava . ^[20]
• Su densidad es infinitamente diferenciable , de hecho supersuave , de orden 2. ^[21]

Además, la densidad de la distribución normal estándar (es decir, y ) también tiene las siguientes propiedades: ${\textstyle \varphi }$ ${\textstyle \mu =0}$ ${\textstyle \sigma =1}$

• Su primera derivada es ${\textstyle \varphi '(x)=-x\varphi (x).}$
• Su segunda derivada es ${\textstyle \varphi ''(x)=(x^{2}-1)\varphi (x)}$
• De manera más general, su derivada n- ésima es donde es el n -ésimo polinomio de Hermite (probabilista) . ^[22] ${\textstyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),}$ ${\textstyle \operatorname {He} _{n}(x)}$
• La probabilidad de que una variable distribuida normalmente con y conocidos esté en un conjunto particular, se puede calcular utilizando el hecho de que la fracción tiene una distribución normal estándar. ${\textstyle X}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle Z=(X-\mu )/\sigma }$

Momentos

Los momentos simples y absolutos de una variable son los valores esperados de y , respectivamente. Si el valor esperado de es cero, estos parámetros se denominan momentos centrales; de lo contrario, estos parámetros se denominan momentos no centrales. Por lo general, solo nos interesan los momentos con orden entero . ${\textstyle X}$ ${\textstyle X^{p}}$ ${\textstyle |X|^{p}}$ ${\textstyle \mu }$ ${\textstyle X}$ ${\textstyle \ p}$

Si tiene una distribución normal, los momentos no centrales existen y son finitos para cualquier número cuya parte real sea mayor que −1. Para cualquier entero no negativo , los momentos centrales simples son: ^[23] Aquí denota el factorial doble , es decir, el producto de todos los números de a 1 que tienen la misma paridad que ${\textstyle X}$ ${\textstyle p}$ ${\textstyle p}$ $${\displaystyle \operatorname {E} \left[(X-\mu )^{p}\right]={\begin{cases}0&{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&{\text{if }}p{\text{ is even.}}\end{cases}}}$$ ${\textstyle n!!}$ ${\textstyle n}$ ${\textstyle n.}$

Los momentos absolutos centrales coinciden con los momentos simples para todos los órdenes pares, pero son distintos de cero para los órdenes impares. Para cualquier entero no negativo ${\textstyle p,}$

$${\displaystyle {\begin{aligned}\operatorname {E} \left[|X-\mu |^{p}\right]&=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{if }}p{\text{ is odd}}\\1&{\text{if }}p{\text{ is even}}\end{cases}}\\&=\sigma ^{p}\cdot {\frac {2^{p/2}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}.\end{aligned}}}$$La última fórmula es válida también para cualquier número no entero. Cuando la media de los momentos simples y absolutos se pueden expresar en términos de funciones hipergeométricas confluentes y ^[24] ${\textstyle p>-1.}$ ${\textstyle \mu \neq 0,}$ ${\textstyle {}_{1}F_{1}}$ ${\textstyle U.}$

$${\displaystyle {\begin{aligned}\operatorname {E} \left[X^{p}\right]&=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}U\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),\\\operatorname {E} \left[|X|^{p}\right]&=\sigma ^{p}\cdot 2^{p/2}{\frac {\Gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}F_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).\end{aligned}}}$$

Estas expresiones siguen siendo válidas incluso si no es un número entero. Véase también polinomios de Hermite generalizados . ${\textstyle p}$

La esperanza de condicionada al evento que se encuentra en un intervalo está dada por donde y respectivamente son la densidad y la función de distribución acumulada de . Esto se conoce como el coeficiente de Mills inverso . Tenga en cuenta que anteriormente, se utiliza la densidad de en lugar de la densidad normal estándar como en el coeficiente de Mills inverso, por lo que aquí tenemos en lugar de . ${\textstyle X}$ ${\textstyle X}$ ${\textstyle [a,b]}$ $${\displaystyle \operatorname {E} \left[X\mid a<X<b\right]=\mu -\sigma ^{2}{\frac {f(b)-f(a)}{F(b)-F(a)}}\,,}$$ ${\textstyle f}$ ${\textstyle F}$ ${\textstyle X}$ ${\textstyle b=\infty }$ ${\textstyle f}$ ${\textstyle X}$ ${\textstyle \sigma ^{2}}$ ${\textstyle \sigma }$

Transformada de Fourier y función característica

La transformada de Fourier de una densidad normal con media y varianza es ^[25] ${\textstyle f}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$

$${\displaystyle {\hat {f}}(t)=\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx=e^{-i\mu t}e^{-{\frac {1}{2}}(\sigma t)^{2}}\,,}$$

donde es la unidad imaginaria . Si la media , el primer factor es 1, y la transformada de Fourier es, además de un factor constante, una densidad normal en el dominio de la frecuencia , con media 0 y varianza . En particular, la distribución normal estándar es una función propia de la transformada de Fourier. ${\textstyle i}$ ${\textstyle \mu =0}$ ${\textstyle 1/\sigma ^{2}}$ ${\textstyle \varphi }$

En teoría de la probabilidad, la transformada de Fourier de la distribución de probabilidad de una variable aleatoria de valor real está estrechamente relacionada con la función característica de esa variable, que se define como el valor esperado de , en función de la variable real (el parámetro de frecuencia de la transformada de Fourier). Esta definición se puede extender analíticamente a una variable de valor complejo . ^[26] La relación entre ambas es: ${\textstyle X}$ ${\textstyle \varphi _{X}(t)}$ ${\textstyle e^{itX}}$ ${\textstyle t}$ ${\textstyle t}$ $${\displaystyle \varphi _{X}(t)={\hat {f}}(-t)\,.}$$

Funciones generadoras de momentos y cumulantes

La función generadora de momentos de una variable aleatoria real es el valor esperado de , en función del parámetro real . Para una distribución normal con densidad , media y varianza , la función generadora de momentos existe y es igual a ${\textstyle X}$ ${\textstyle e^{tX}}$ ${\textstyle t}$ ${\textstyle f}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$

$${\displaystyle M(t)=\operatorname {E} \left[e^{tX}\right]={\hat {f}}(it)=e^{\mu t}e^{\sigma ^{2}t^{2}/2}\,.}$$Para cualquier ⁠ ⁠ ${\displaystyle k}$ , el coeficiente de ⁠ ⁠ ${\displaystyle t^{k}/k!}$ en la función generadora de momentos (expresada como una serie de potencia exponencial en ⁠ ⁠ ${\displaystyle t}$ ) es el valor esperado de la distribución normal ⁠ ⁠ ${\displaystyle E[X^{k}]}$ .

La función generadora cumulante es el logaritmo de la función generadora de momentos, es decir

$${\displaystyle g(t)=\ln M(t)=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}\,.}$$

Los coeficientes de esta serie de potencias exponenciales definen los cumulantes, pero como se trata de un polinomio cuadrático en ⁠ ⁠ ${\displaystyle t}$ , solo los dos primeros cumulantes son distintos de cero, es decir, la media  y la varianza  ⁠ ⁠ . ${\textstyle \mu }$ ${\displaystyle \sigma ^{2}}$

Algunos autores prefieren trabajar con E[ e ^itX ] = e ^{iμt − σ 2 t 2 /2} y ln E[ e ^itX ] = iμt − ⁠1/2⁠ σ2t2 .^​​^​

Operador y clase Stein

Dentro del método de Stein, el operador de Stein y la clase de una variable aleatoria son y la clase de todas las funciones absolutamente continuas . ${\textstyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ ${\textstyle {\mathcal {A}}f(x)=\sigma ^{2}f'(x)-(x-\mu )f(x)}$ ${\textstyle {\mathcal {F}}}$ ${\textstyle f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f'(X)|]<\infty }$

Límite de varianza cero

En el límite cuando tiende a cero, la densidad de probabilidad tiende eventualmente a cero en cualquier , pero crece sin límite si , mientras que su integral permanece igual a 1. Por lo tanto, la distribución normal no puede definirse como una función ordinaria cuando . ${\textstyle \sigma ^{2}}$ ${\textstyle f(x)}$ ${\textstyle x\neq \mu }$ ${\textstyle x=\mu }$ ${\textstyle \sigma ^{2}=0}$

Sin embargo, se puede definir la distribución normal con varianza cero como una función generalizada ; específicamente, como una función delta de Dirac traducida por la media , es decir Su función de distribución acumulativa es entonces la función escalón de Heaviside traducida por la media , es decir ${\textstyle \delta }$ ${\textstyle \mu }$ ${\textstyle f(x)=\delta (x-\mu ).}$ ${\textstyle \mu }$ $${\displaystyle F(x)={\begin{cases}0&{\text{if }}x<\mu \\1&{\text{if }}x\geq \mu \,.\end{cases}}}$$

Entropía máxima

De todas las distribuciones de probabilidad sobre los números reales con una media finita especificada y una varianza finita  , la distribución normal es la que tiene una entropía máxima . ^[27] Para ver esto, sea una variable aleatoria continua con densidad de probabilidad . La entropía de se define como ^{[28] [29] [30]} ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle N(\mu ,\sigma ^{2})}$ ${\textstyle X}$ ${\textstyle f(x)}$ ${\textstyle X}$ $${\displaystyle H(X)=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx\,,}$$

donde se entiende que es cero siempre que . Esta función se puede maximizar, sujeta a las restricciones de que la distribución esté correctamente normalizada y tenga una media y varianza especificadas, mediante el uso del cálculo variacional . Se define una función con tres multiplicadores de Lagrange : ${\textstyle f(x)\log f(x)}$ ${\textstyle f(x)=0}$

$${\displaystyle L=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx-\lambda _{0}\left(1-\int _{-\infty }^{\infty }f(x)\,dx\right)-\lambda _{1}\left(\mu -\int _{-\infty }^{\infty }f(x)x\,dx\right)-\lambda _{2}\left(\sigma ^{2}-\int _{-\infty }^{\infty }f(x)(x-\mu )^{2}\,dx\right)\,.}$$

Con máxima entropía, una pequeña variación de producirá una variación de que es igual a 0: ${\textstyle \delta f(x)}$ ${\textstyle f(x)}$ ${\textstyle \delta L}$ ${\textstyle L}$

$${\displaystyle 0=\delta L=\int _{-\infty }^{\infty }\delta f(x)\left(-\ln f(x)-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,dx\,.}$$

Dado que esto debe ser válido para cualquier α pequeño , el factor que se multiplica debe ser cero, y al resolver se obtiene: ${\textstyle \delta f(x)}$ ${\textstyle \delta f(x)}$ ${\textstyle f(x)}$

$${\displaystyle f(x)=\exp \left(-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,.}$$

Las restricciones de Lagrange que están correctamente normalizadas y tienen la media y varianza especificadas se satisfacen si y solo si , , y se eligen de modo que La entropía de una distribución normal es igual a que es independiente de la media . ${\textstyle f(x)}$ ${\textstyle \lambda _{0}}$ ${\textstyle \lambda _{1}}$ ${\textstyle \lambda _{2}}$ $${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}$$ ${\textstyle X\sim N(\mu ,\sigma ^{2})}$ $${\displaystyle H(X)={\tfrac {1}{2}}(1+\ln 2\sigma ^{2}\pi )\,,}$$ ${\textstyle \mu }$

Otras propiedades

1. Si la función característica de alguna variable aleatoria es de la forma en un entorno de cero, donde es un polinomio , entonces el teorema de Marcinkiewicz (llamado así por Józef Marcinkiewicz ) afirma que puede ser como máximo un polinomio cuadrático y, por lo tanto, es una variable aleatoria normal. ^{[31] La consecuencia de este resultado es que la distribución normal es la única distribución con un número finito (dos) de} cumulantes distintos de cero . ${\textstyle \phi _{X}}$ ${\textstyle X}$ ${\textstyle \phi _{X}(t)=\exp Q(t)}$ ${\textstyle Q(t)}$ ${\textstyle Q}$ ${\textstyle X}$
2. Si y son conjuntamente normales y no correlacionados , entonces son independientes . El requisito de que y sean conjuntamente normales es esencial; sin él la propiedad no se cumple. ^{[32] [33] [prueba]} Para las variables aleatorias no normales, la no correlación no implica independencia. ${\textstyle X}$ ${\textstyle Y}$ ${\textstyle X}$ ${\textstyle Y}$
3. La divergencia de Kullback-Leibler de una distribución normal respecto de otra está dada por: ^[34] La distancia de Hellinger entre las mismas distribuciones es igual a ${\textstyle X_{1}\sim N(\mu _{1},\sigma _{1}^{2})}$ ${\textstyle X_{2}\sim N(\mu _{2},\sigma _{2}^{2})}$ $${\displaystyle D_{\mathrm {KL} }(X_{1}\parallel X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)}$$ $${\displaystyle H^{2}(X_{1},X_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\exp \left(-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)}$$
4. La matriz de información de Fisher para una distribución normal con respecto a y es diagonal y toma la forma ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ $${\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&0\\0&{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}}$$
5. La distribución conjugada anterior de la media de una distribución normal es otra distribución normal. ^[35] Específicamente, si son iid y la distribución anterior es , entonces la distribución posterior para el estimador de será ${\textstyle x_{1},\ldots ,x_{n}}$ ${\textstyle \sim N(\mu ,\sigma ^{2})}$ ${\textstyle \mu \sim N(\mu _{0},\sigma _{0}^{2})}$ ${\textstyle \mu }$ $${\displaystyle \mu \mid x_{1},\ldots ,x_{n}\sim {\mathcal {N}}\left({\frac {{\frac {\sigma ^{2}}{n}}\mu _{0}+\sigma _{0}^{2}{\bar {x}}}{{\frac {\sigma ^{2}}{n}}+\sigma _{0}^{2}}},\left({\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}\right)^{-1}\right)}$$
6. La familia de distribuciones normales no solo forma una familia exponencial (EF), sino que de hecho forma una familia exponencial natural (NEF) con función de varianza cuadrática ( NEF-QVF ). Muchas propiedades de las distribuciones normales se generalizan a propiedades de las distribuciones NEF-QVF, distribuciones NEF o distribuciones EF en general. Las distribuciones NEF-QVF comprenden 6 familias, incluidas las distribuciones Poisson, Gamma, binomial y binomial negativa, mientras que muchas de las familias comunes estudiadas en probabilidad y estadística son NEF o EF.
7. En geometría de la información , la familia de distribuciones normales forma una variedad estadística con curvatura constante . La misma familia es plana con respecto a las conexiones (±1) y . ^[36] ${\textstyle -1}$ ${\textstyle \nabla ^{(e)}}$ ${\textstyle \nabla ^{(m)}}$
8. Si se distribuyen según , entonces . Nótese que no hay ningún supuesto de independencia. ^[37] ${\textstyle X_{1},\dots ,X_{n}}$ ${\textstyle N(0,\sigma ^{2})}$ ${\textstyle E[\max _{i}X_{i}]\leq \sigma {\sqrt {2\ln n}}}$

Distribuciones relacionadas

Teorema del límite central

A medida que aumenta el número de eventos discretos, la función comienza a parecerse a una distribución normal.

Comparación de funciones de densidad de probabilidad para la suma de dados de 6 caras para mostrar su convergencia a una distribución normal con un aumento de , de acuerdo con el teorema del límite central. En el gráfico inferior derecho, los perfiles suavizados de los gráficos anteriores se reescalan, se superponen y se comparan con una distribución normal (curva negra). ${\textstyle p(k)}$ ${\textstyle n}$ ${\textstyle na}$

El teorema del límite central establece que, en determinadas condiciones (bastante comunes), la suma de muchas variables aleatorias tendrá una distribución aproximadamente normal. Más específicamente, donde son variables aleatorias independientes e idénticamente distribuidas con la misma distribución arbitraria, media y varianza cero y su media está escalada por Entonces, a medida que aumenta, la distribución de probabilidad de tenderá a la distribución normal con media y varianza cero . ${\textstyle X_{1},\ldots ,X_{n}}$ ${\textstyle \sigma ^{2}}$ ${\textstyle Z}$ ${\textstyle {\sqrt {n}}}$ $${\displaystyle Z={\sqrt {n}}\left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)}$$ ${\textstyle n}$ ${\textstyle Z}$ ${\textstyle \sigma ^{2}}$

El teorema puede extenderse a variables que no son independientes y/o no están distribuidas idénticamente si se imponen ciertas restricciones al grado de dependencia y a los momentos de las distribuciones. ${\textstyle (X_{i})}$

Muchas estadísticas de prueba , puntuaciones y estimadores que se encuentran en la práctica contienen sumas de ciertas variables aleatorias, e incluso más estimadores pueden representarse como sumas de variables aleatorias mediante el uso de funciones de influencia . El teorema del límite central implica que esos parámetros estadísticos tendrán distribuciones asintóticamente normales.

El teorema del límite central también implica que ciertas distribuciones pueden aproximarse mediante la distribución normal, por ejemplo:

• La distribución binomial es aproximadamente normal con media y varianza grandes y no demasiado cercanas a 0 o 1. ${\textstyle B(n,p)}$ ${\textstyle np}$ ${\textstyle np(1-p)}$ ${\textstyle n}$ ${\textstyle p}$
• La distribución de Poisson con parámetro es aproximadamente normal con media y varianza , para valores grandes de . ^[38] ${\textstyle \lambda }$ ${\textstyle \lambda }$ ${\textstyle \lambda }$ ${\textstyle \lambda }$
• La distribución de chi-cuadrado es aproximadamente normal con media y varianza , para valores grandes . ${\textstyle \chi ^{2}(k)}$ ${\textstyle k}$ ${\textstyle 2k}$ ${\textstyle k}$
• La distribución t de Student es aproximadamente normal con media 0 y varianza 1 cuando es grande. ${\textstyle t(\nu )}$ ${\textstyle \nu }$

La precisión de estas aproximaciones depende del propósito para el que se necesitan y de la tasa de convergencia a la distribución normal. Por lo general, estas aproximaciones son menos precisas en los extremos de la distribución.

El teorema de Berry-Esseen proporciona un límite superior general para el error de aproximación en el teorema del límite central y las expansiones de Edgeworth proporcionan mejoras en la aproximación .

Este teorema también se puede utilizar para justificar la modelización de la suma de muchas fuentes de ruido uniformes como ruido gaussiano . Véase AWGN .

Operaciones y funciones de variables normales

a: Densidad de probabilidad de una función de una variable normal con y . b: Densidad de probabilidad de una función de dos variables normales y , donde , , , , y . c: Mapa de calor de la densidad de probabilidad conjunta de dos funciones de dos variables normales correlacionadas y , donde , , , , y . d: Densidad de probabilidad de una función de 4 variables normales estándar iid. Estas se calculan mediante el método numérico de trazado de rayos. ^[39] ${\textstyle \cos x^{2}}$ ${\textstyle x}$ ${\textstyle \mu =-2}$ ${\textstyle \sigma =3}$ ${\textstyle x^{y}}$ ${\textstyle x}$ ${\textstyle y}$ ${\textstyle \mu _{x}=1}$ ${\textstyle \mu _{y}=2}$ ${\textstyle \sigma _{x}=0.1}$ ${\textstyle \sigma _{y}=0.2}$ ${\textstyle \rho _{xy}=0.8}$ ${\textstyle x}$ ${\textstyle y}$ ${\textstyle \mu _{x}=-2}$ ${\textstyle \mu _{y}=5}$ ${\textstyle \sigma _{x}^{2}=10}$ ${\textstyle \sigma _{y}^{2}=20}$ ${\textstyle \rho _{xy}=0.495}$ ${\textstyle \sum _{i=1}^{4}\vert x_{i}\vert }$

La densidad de probabilidad , la distribución acumulativa y la distribución acumulativa inversa de cualquier función de una o más variables normales independientes o correlacionadas se pueden calcular con el método numérico de trazado de rayos ^[39] (código Matlab). En las siguientes secciones, analizamos algunos casos especiales.

Operaciones sobre una sola variable normal

Si se distribuye normalmente con media y varianza , entonces ${\textstyle X}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$

• ${\textstyle aX+b}$, para cualquier número real y , también se distribuye normalmente, con media y varianza . Es decir, la familia de distribuciones normales está cerrada ante transformaciones lineales . ${\textstyle a}$ ${\textstyle b}$ ${\textstyle a\mu +b}$ ${\textstyle a^{2}\sigma ^{2}}$
• La exponencial de se distribuye log-normalmente : . ${\textstyle X}$ ${\textstyle e^{X}\sim \ln(N(\mu ,\sigma ^{2}))}$
• La sigmoide estándar de tiene una distribución logit-normal : . ${\textstyle X}$ ${\textstyle \sigma (X)\sim P({\mathcal {N}}(\mu ,\,\sigma ^{2}))}$
• El valor absoluto de tiene una distribución normal plegada : . Si esto se conoce como distribución seminormal . ${\textstyle X}$ ${\textstyle {\left|X\right|\sim N_{f}(\mu ,\sigma ^{2})}}$ ${\textstyle \mu =0}$
• El valor absoluto de los residuos normalizados, , tiene distribución chi con un grado de libertad: . ${\textstyle |X-\mu |/\sigma }$ ${\textstyle |X-\mu |/\sigma \sim \chi _{1}}$
• El cuadrado de tiene una distribución chi-cuadrado no central con un grado de libertad: . Si , la distribución se denomina simplemente chi-cuadrado . ${\textstyle X/\sigma }$ ${\textstyle X^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})}$ ${\textstyle \mu =0}$
• La verosimilitud logarítmica de una variable normal es simplemente el logaritmo de su función de densidad de probabilidad : dado que se trata de un cuadrado escalado y desplazado de una variable normal estándar, se distribuye como una variable chi-cuadrado escalada y desplazada. ${\textstyle x}$ $${\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).}$$
• La distribución de la variable restringida a un intervalo se denomina distribución normal truncada . ${\textstyle X}$ ${\textstyle [a,b]}$
• ${\textstyle (X-\mu )^{-2}}$tiene una distribución de Lévy con ubicación 0 y escala . ${\textstyle \sigma ^{-2}}$

Operaciones sobre dos variables normales independientes

• Si y son dos variables aleatorias normales independientes , con medias , y varianzas , , entonces su suma también estará distribuida normalmente, ^[prueba] con media y varianza . ${\textstyle X_{1}}$ ${\textstyle X_{2}}$ ${\textstyle \mu _{1}}$ ${\textstyle \mu _{2}}$ ${\textstyle \sigma _{1}^{2}}$ ${\textstyle \sigma _{2}^{2}}$ ${\textstyle X_{1}+X_{2}}$ ${\textstyle \mu _{1}+\mu _{2}}$ ${\textstyle \sigma _{1}^{2}+\sigma _{2}^{2}}$
• En particular, si y son desviaciones normales independientes con media y varianza cero , entonces y también son independientes y se distribuyen normalmente, con media y varianza cero . Este es un caso especial de la identidad de polarización . ^[40] ${\textstyle X}$ ${\textstyle Y}$ ${\textstyle \sigma ^{2}}$ ${\textstyle X+Y}$ ${\textstyle X-Y}$ ${\textstyle 2\sigma ^{2}}$
• Si , son dos desviaciones normales independientes con media y varianza , y , son números reales arbitrarios, entonces la variable también se distribuye normalmente con media y varianza . De ello se deduce que la distribución normal es estable (con exponente ). ${\textstyle X_{1}}$ ${\textstyle X_{2}}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle a}$ ${\textstyle b}$ $${\displaystyle X_{3}={\frac {aX_{1}+bX_{2}-(a+b)\mu }{\sqrt {a^{2}+b^{2}}}}+\mu }$$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle \alpha =2}$
• Si , son distribuciones normales, entonces su media geométrica normalizada es una distribución normal con y (ver aquí para una visualización). ${\textstyle X_{k}\sim {\mathcal {N}}(m_{k},\sigma _{k}^{2})}$ ${\textstyle k\in \{0,1\}}$ ${\textstyle {\frac {1}{\int _{\mathbb {R} ^{n}}X_{0}^{\alpha }(x)X_{1}^{1-\alpha }(x)\,{\text{d}}x}}X_{0}^{\alpha }X_{1}^{1-\alpha }}$ ${\textstyle {\mathcal {N}}(m_{\alpha },\sigma _{\alpha }^{2})}$ ${\textstyle m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}$ ${\textstyle \sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}$

Operaciones sobre dos variables normales estándar independientes

Si y son dos variables aleatorias normales estándar independientes con media 0 y varianza 1, entonces ${\textstyle X_{1}}$ ${\textstyle X_{2}}$

• Su suma y diferencia se distribuyen normalmente con media cero y varianza dos: . ${\textstyle X_{1}\pm X_{2}\sim {\mathcal {N}}(0,2)}$
• Su producto sigue la distribución de producto ^[41] con función de densidad donde es la función de Bessel modificada de segundo tipo . Esta distribución es simétrica alrededor de cero, ilimitada en , y tiene la función característica . ${\textstyle Z=X_{1}X_{2}}$ ${\textstyle f_{Z}(z)=\pi ^{-1}K_{0}(|z|)}$ ${\textstyle K_{0}}$ ${\textstyle z=0}$ ${\textstyle \phi _{Z}(t)=(1+t^{2})^{-1/2}}$
• Su relación sigue la distribución estándar de Cauchy : . ${\textstyle X_{1}/X_{2}\sim \operatorname {Cauchy} (0,1)}$
• Su norma euclidiana tiene la distribución de Rayleigh . ${\textstyle {\sqrt {X_{1}^{2}+X_{2}^{2}}}}$

Operaciones sobre múltiples variables normales independientes

• Cualquier combinación lineal de desviaciones normales independientes es una desviación normal.
• Si son variables aleatorias normales estándar independientes, entonces la suma de sus cuadrados tiene la distribución de chi-cuadrado con grados de libertad. ${\textstyle X_{1},X_{2},\ldots ,X_{n}}$ ${\textstyle n}$ $${\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}\sim \chi _{n}^{2}.}$$
• Si son variables aleatorias independientes distribuidas normalmente con medias y varianzas , entonces su media muestral es independiente de la desviación estándar muestral , ^[42] lo que se puede demostrar utilizando el teorema de Basu o el teorema de Cochran . ^[43] La relación de estas dos cantidades tendrá la distribución t de Student con grados de libertad: ${\textstyle X_{1},X_{2},\ldots ,X_{n}}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle n-1}$ $${\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}\right]}}}\sim t_{n-1}.}$$
• Si , son variables aleatorias normales estándar independientes, entonces la relación de sus sumas de cuadrados normalizadas tendrá la distribución F con ( n , m ) grados de libertad: ^[44] ${\textstyle X_{1},X_{2},\ldots ,X_{n}}$ ${\textstyle Y_{1},Y_{2},\ldots ,Y_{m}}$ $${\displaystyle F={\frac {\left(X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{m}^{2}\right)/m}}\sim F_{n,m}.}$$

Operaciones sobre múltiples variables normales correlacionadas

• Una forma cuadrática de un vector normal, es decir, una función cuadrática de múltiples variables normales independientes o correlacionadas, es una variable de chi-cuadrado generalizada . ${\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c}$

Operaciones sobre la función de densidad

La distribución normal dividida se define de manera más directa en términos de unir secciones escaladas de las funciones de densidad de diferentes distribuciones normales y reescalar la densidad para integrarla en una sola. La distribución normal truncada resulta de reescalar una sección de una sola función de densidad.

Divisibilidad infinita y teorema de Cramér

Para cualquier entero positivo , cualquier distribución normal con media y varianza es la distribución de la suma de desviaciones normales independientes, cada una con media y varianza . Esta propiedad se llama divisibilidad infinita . ^[45] ${\textstyle {\text{n}}}$ ${\textstyle \mu }$ ${\textstyle \sigma ^{2}}$ ${\textstyle {\text{n}}}$ ${\textstyle {\frac {\mu }{n}}}$ ${\textstyle {\frac {\sigma ^{2}}{n}}}$

Por el contrario, si y son variables aleatorias independientes y su suma tiene una distribución normal, entonces tanto como deben ser desviaciones normales. ^[46] ${\textstyle X_{1}}$ ${\textstyle X_{2}}$ ${\textstyle X_{1}+X_{2}}$ ${\textstyle X_{1}}$ ${\textstyle X_{2}}$

Este resultado se conoce como teorema de descomposición de Cramér y equivale a decir que la convolución de dos distribuciones es normal si y solo si ambas son normales. El teorema de Cramér implica que una combinación lineal de variables independientes no gaussianas nunca tendrá una distribución exactamente normal, aunque puede aproximarse a ella de forma arbitraria. ^[31]

El teorema de Kac-Bernstein

The Kac–Bernstein theorem states that if ${\textstyle X}$ and ${\textstyle Y}$ are independent and ${\textstyle X+Y}$ and ${\textstyle X-Y}$ are also independent, then both X and Y must necessarily have normal distributions.^[47][48]

More generally, if ${\textstyle X_{1},\ldots ,X_{n}}$ are independent random variables, then two distinct linear combinations ${\textstyle \sum {a_{k}X_{k}}}$ and ${\textstyle \sum {b_{k}X_{k}}}$will be independent if and only if all ${\textstyle X_{k}}$ are normal and ${\textstyle \sum {a_{k}b_{k}\sigma _{k}^{2}=0}}$, where ${\textstyle \sigma _{k}^{2}}$ denotes the variance of ${\textstyle X_{k}}$.^[47]

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.

• The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R^k is multivariate-normally distributed if any linear combination of its components Σ^k
  _j=1a_j X_j has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
• Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0
• Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ C^k is said to be normal if both its real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.
• Matrix normal distribution describes the case of normally distributed matrices.
• Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space H, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element h ∈ H is said to be normal if for any constant a ∈ H the scalar product (a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:
  • Brownian motion,
  • Brownian bridge,
  • Ornstein–Uhlenbeck process.
• Gaussian q-distribution is an abstract mathematical construction that represents a q-analogue of the normal distribution.
• the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution above.
• The Kaniadakis κ-Gaussian distribution is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable X has a two-piece normal distribution if it has a distribution

$${\displaystyle f_{X}(x)=N(\mu ,\sigma _{1}^{2}){\text{ if }}x\leq \mu }$$ $${\displaystyle f_{X}(x)=N(\mu ,\sigma _{2}^{2}){\text{ if }}x\geq \mu }$$

where μ is the mean and σ₁² and σ₂² are the variances of the distribution to the left and right of the mean respectively.

The mean, variance and third central moment of this distribution have been determined^[49]

$${\displaystyle \operatorname {E} (X)=\mu +{\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})}$$ $${\displaystyle \operatorname {V} (X)=\left(1-{\frac {2}{\pi }}\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}}$$ $${\displaystyle \operatorname {T} (X)={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}\right]}$$

where E(X), V(X) and T(X) are the mean, variance, and third central moment respectively.

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:

• Pearson distribution — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
• The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample ${\textstyle (x_{1},\ldots ,x_{n})}$ from a normal ${\textstyle {\mathcal {N}}(\mu ,\sigma ^{2})}$ population we would like to learn the approximate values of parameters ${\textstyle \mu }$ and ${\textstyle \sigma ^{2}}$. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function: $${\displaystyle \ln {\mathcal {L}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}$$Taking derivatives with respect to ${\textstyle \mu }$ and ${\textstyle \sigma ^{2}}$ and solving the resulting system of first order conditions yields the maximum likelihood estimates: $${\displaystyle {\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}$$

Then ${\textstyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})}$ is as follows:

$${\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})=(-n/2)[\ln(2\pi {\hat {\sigma }}^{2})+1]}$$

Sample mean

Estimator ${\displaystyle \textstyle {\hat {\mu }}}$ is called the sample mean, since it is the arithmetic mean of all observations. The statistic ${\displaystyle \textstyle {\overline {x}}}$ is complete and sufficient for ${\textstyle \mu }$, and therefore by the Lehmann–Scheffé theorem, ${\displaystyle \textstyle {\hat {\mu }}}$ is the uniformly minimum variance unbiased (UMVU) estimator.^[50] In finite samples it is distributed normally: $${\displaystyle {\hat {\mu }}\sim {\mathcal {N}}(\mu ,\sigma ^{2}/n).}$$The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix ${\displaystyle \textstyle {\mathcal {I}}^{-1}}$. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of ${\displaystyle \textstyle {\hat {\mu }}}$ is proportional to ${\displaystyle \textstyle 1/{\sqrt {n}}}$, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.

From the standpoint of the asymptotic theory, ${\displaystyle \textstyle {\hat {\mu }}}$ is consistent, that is, it converges in probability to ${\textstyle \mu }$ as ${\textstyle n\rightarrow \infty }$. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples: $${\displaystyle {\sqrt {n}}({\hat {\mu }}-\mu )\,\xrightarrow {d} \,{\mathcal {N}}(0,\sigma ^{2}).}$$

Sample variance

The estimator ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ is called the sample variance, since it is the variance of the sample ( ${\textstyle (x_{1},\ldots ,x_{n})}$). In practice, another estimator is often used instead of the ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$. This other estimator is denoted ${\textstyle s^{2}}$, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root ${\textstyle s}$ is called the sample standard deviation. The estimator ${\textstyle s^{2}}$ differs from ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ by having (n − 1) instead of n in the denominator (the so-called Bessel's correction): $${\displaystyle s^{2}={\frac {n}{n-1}}{\hat {\sigma }}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}$$The difference between ${\textstyle s^{2}}$ and ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ becomes negligibly small for large n's. In finite samples however, the motivation behind the use of ${\textstyle s^{2}}$ is that it is an unbiased estimator of the underlying parameter ${\textstyle \sigma ^{2}}$, whereas ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ is biased. Also, by the Lehmann–Scheffé theorem the estimator ${\textstyle s^{2}}$ is uniformly minimum variance unbiased (UMVU),^[50] which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ is better than the ${\textstyle s^{2}}$ in terms of the mean squared error (MSE) criterion. In finite samples both ${\textstyle s^{2}}$ and ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ have scaled chi-squared distribution with (n − 1) degrees of freedom: $${\displaystyle s^{2}\sim {\frac {\sigma ^{2}}{n-1}}\cdot \chi _{n-1}^{2},\qquad {\hat {\sigma }}^{2}\sim {\frac {\sigma ^{2}}{n}}\cdot \chi _{n-1}^{2}.}$$The first of these expressions shows that the variance of ${\textstyle s^{2}}$ is equal to ${\textstyle 2\sigma ^{4}/(n-1)}$, which is slightly greater than the σσ-element of the inverse Fisher information matrix ${\displaystyle \textstyle {\mathcal {I}}^{-1}}$. Thus, ${\textstyle s^{2}}$ is not an efficient estimator for ${\textstyle \sigma ^{2}}$, and moreover, since ${\textstyle s^{2}}$ is UMVU, we can conclude that the finite-sample efficient estimator for ${\textstyle \sigma ^{2}}$ does not exist.

Applying the asymptotic theory, both estimators ${\textstyle s^{2}}$ and ${\displaystyle \textstyle {\hat {\sigma }}^{2}}$ are consistent, that is they converge in probability to ${\textstyle \sigma ^{2}}$ as the sample size ${\textstyle n\rightarrow \infty }$. The two estimators are also both asymptotically normal: $${\displaystyle {\sqrt {n}}({\hat {\sigma }}^{2}-\sigma ^{2})\simeq {\sqrt {n}}(s^{2}-\sigma ^{2})\,\xrightarrow {d} \,{\mathcal {N}}(0,2\sigma ^{4}).}$$In particular, both estimators are asymptotically efficient for ${\textstyle \sigma ^{2}}$.

Confidence intervals

By Cochran's theorem, for normal distributions the sample mean ${\displaystyle \textstyle {\hat {\mu }}}$ and the sample variance s² are independent, which means there can be no gain in considering their joint distribution. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between ${\displaystyle \textstyle {\hat {\mu }}}$ and s can be employed to construct the so-called t-statistic: $${\displaystyle t={\frac {{\hat {\mu }}-\mu }{s/{\sqrt {n}}}}={\frac {{\overline {x}}-\mu }{\sqrt {{\frac {1}{n(n-1)}}\sum (x_{i}-{\overline {x}})^{2}}}}\sim t_{n-1}}$$This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;^[51] similarly, inverting the χ² distribution of the statistic s² will give us the confidence interval for σ²:^[52] $${\displaystyle \mu \in \left[{\hat {\mu }}-{\frac {t_{n-1,1-\alpha /2}}{\sqrt {n}}}s,\,{\hat {\mu }}+{\frac {t_{n-1,1-\alpha /2}}{\sqrt {n}}}s\right],}$$ $${\displaystyle \sigma ^{2}\in \left[{\frac {n-1}{\chi _{n-1,1-\alpha /2}^{2}}}s^{2},\,{\frac {n-1}{\chi _{n-1,\alpha /2}^{2}}}s^{2}\right],}$$where t_k,p and χ 2
k,p  are the pth quantiles of the t- and χ²-distributions respectively. These confidence intervals are of the confidence level 1 − α, meaning that the true values μ and σ² fall outside of these intervals with probability (or significance level) α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The confidence interval for σ can be found by taking the square root of the interval bounds for σ².

Approximate formulas can be derived from the asymptotic distributions of ${\displaystyle \textstyle {\hat {\mu }}}$ and s²: $${\displaystyle \mu \in \left[{\hat {\mu }}-{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s,\,{\hat {\mu }}+{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s\right],}$$ $${\displaystyle \sigma ^{2}\in \left[s^{2}-{\sqrt {2}}{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s^{2},\,s^{2}+{\sqrt {2}}{\frac {|z_{\alpha /2}|}{\sqrt {n}}}s^{2}\right],}$$The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles z_α/2 do not depend on n. In particular, the most popular value of α = 5%, results in |z_0.025| = 1.96.

Normality tests

Normality tests assess the likelihood that the given data set {x₁, ..., x_n} comes from a normal distribution. Typically the null hypothesis H₀ is that the observations are distributed normally with unspecified mean μ and variance σ², versus the alternative H_a that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.

• Q–Q plot, also known as normal probability plot or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(p_k), x_(k)), where plotting points p_k are equal to p_k = (k − α)/(n + 1 − 2α) and α is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
• P–P plot – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(z_(k)), p_k), where ${\textstyle \textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}}$. For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).

Goodness-of-fit tests:

Moment-based tests:

• D'Agostino's K-squared test
• Jarque–Bera test
• Shapiro–Wilk test: This is based on the fact that the line in the Q–Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.

Tests based on the empirical distribution function:

• Anderson–Darling test
• Lilliefors test (an adaptation of the Kolmogorov–Smirnov test)

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:

• Either the mean, or the variance, or neither, may be considered a fixed quantity.
• When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
• Both univariate and multivariate cases need to be considered.
• Either conjugate or improper prior distributions may be placed on the unknown variables.
• An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior article.

Sum of two quadratics

Scalar form

The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

$${\displaystyle a(x-y)^{2}+b(x-z)^{2}=(a+b)\left(x-{\frac {ay+bz}{a+b}}\right)^{2}+{\frac {ab}{a+b}}(y-z)^{2}}$$

This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:

1. The factor ${\textstyle {\frac {ay+bz}{a+b}}}$ has the form of a weighted average of y and z.
2. ${\textstyle {\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.}$ This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that ${\textstyle {\frac {ab}{a+b}}}$ is one-half the harmonic mean of a and b.

Vector form

A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size ${\textstyle k\times k}$, then

$${\displaystyle {\begin{aligned}&(\mathbf {y} -\mathbf {x} )'\mathbf {A} (\mathbf {y} -\mathbf {x} )+(\mathbf {x} -\mathbf {z} )'\mathbf {B} (\mathbf {x} -\mathbf {z} )\\={}&(\mathbf {x} -\mathbf {c} )'(\mathbf {A} +\mathbf {B} )(\mathbf {x} -\mathbf {c} )+(\mathbf {y} -\mathbf {z} )'(\mathbf {A} ^{-1}+\mathbf {B} ^{-1})^{-1}(\mathbf {y} -\mathbf {z} )\end{aligned}}}$$

where

$${\displaystyle \mathbf {c} =(\mathbf {A} +\mathbf {B} )^{-1}(\mathbf {A} \mathbf {y} +\mathbf {B} \mathbf {z} )}$$

The form x′ A x is called a quadratic form and is a scalar: $${\displaystyle \mathbf {x} '\mathbf {A} \mathbf {x} =\sum _{i,j}a_{ij}x_{i}x_{j}}$$In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since ${\textstyle x_{i}x_{j}=x_{j}x_{i}}$, only the sum ${\textstyle a_{ij}+a_{ji}}$ matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form ${\textstyle \mathbf {x} '\mathbf {A} \mathbf {y} =\mathbf {y} '\mathbf {A} \mathbf {x} .}$

Sum of differences from the mean

Another useful formula is as follows: $${\displaystyle \sum _{i=1}^{n}(x_{i}-\mu )^{2}=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}}$$where ${\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}$

With known variance

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows ${\textstyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ with known variance σ², the conjugate prior distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if ${\textstyle x\sim {\mathcal {N}}(\mu ,1/\tau )}$ and ${\textstyle \mu \sim {\mathcal {N}}(\mu _{0},1/\tau _{0}),}$ we proceed as follows.

First, the likelihood function is (using the formula above for the sum of differences from the mean):

$${\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\tau )&=\prod _{i=1}^{n}{\sqrt {\frac {\tau }{2\pi }}}\exp \left(-{\frac {1}{2}}\tau (x_{i}-\mu )^{2}\right)\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left(-{\frac {1}{2}}\tau \sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right].\end{aligned}}}$$

Then, we proceed as follows:

$${\displaystyle {\begin{aligned}p(\mu \mid \mathbf {X} )&\propto p(\mathbf {X} \mid \mu )p(\mu )\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}}$$

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean ${\textstyle {\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}}$ and precision ${\textstyle n\tau +\tau _{0}}$, i.e.

$${\displaystyle p(\mu \mid \mathbf {X} )\sim {\mathcal {N}}\left({\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}},{\frac {1}{n\tau +\tau _{0}}}\right)}$$

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

$${\displaystyle {\begin{aligned}\tau _{0}'&=\tau _{0}+n\tau \\[5pt]\mu _{0}'&={\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\\[5pt]{\bar {x}}&={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}$$

That is, to combine n data points with total precision of nτ (or equivalently, total variance of n/σ²) and mean of values ${\textstyle {\bar {x}}}$, derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precision-weighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

$${\displaystyle {\begin{aligned}{\sigma _{0}^{2}}'&={\frac {1}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]\mu _{0}'&={\frac {{\frac {n{\bar {x}}}{\sigma ^{2}}}+{\frac {\mu _{0}}{\sigma _{0}^{2}}}}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]{\bar {x}}&={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}$$

With known mean

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows ${\textstyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

$${\displaystyle p(\sigma ^{2}\mid \nu _{0},\sigma _{0}^{2})={\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\nu _{0}/2}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\propto {\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}}$$

The likelihood function from above, written in terms of the variance, is:

$${\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]\end{aligned}}}$$

where

$${\displaystyle S=\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}$$

Then:

$${\displaystyle {\begin{aligned}p(\sigma ^{2}\mid \mathbf {X} )&\propto p(\mathbf {X} \mid \sigma ^{2})p(\sigma ^{2})\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {S}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+S}{2\sigma ^{2}}}\right]\end{aligned}}}$$

The above is also a scaled inverse chi-squared distribution where

$${\displaystyle {\begin{aligned}\nu _{0}'&=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}\end{aligned}}}$$

or equivalently

$${\displaystyle {\begin{aligned}\nu _{0}'&=\nu _{0}+n\\{\sigma _{0}^{2}}'&={\frac {\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{\nu _{0}+n}}\end{aligned}}}$$

Reparameterizing in terms of an inverse gamma distribution, the result is:

$${\displaystyle {\begin{aligned}\alpha '&=\alpha +{\frac {n}{2}}\\\beta '&=\beta +{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{2}}\end{aligned}}}$$

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows ${\textstyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ with unknown mean μ and unknown variance σ², a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution. Logically, this originates as follows:

1. From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistics computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
2. From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
3. Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
4. To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
5. This suggests that we create a conditional prior of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
6. This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate priors used (an inverse gamma distribution over the variance, and a normal distribution over the mean, conditional on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

$${\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\end{aligned}}}$$

The update equations can be derived, and look as follows:

$${\displaystyle {\begin{aligned}{\bar {x}}&={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\\\mu _{0}'&={\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\\n_{0}'&=n_{0}+n\\\nu _{0}'&=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\end{aligned}}}$$

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for ${\textstyle \nu _{0}'{\sigma _{0}^{2}}'}$ is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Proof

The prior distributions are $${\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})={\frac {1}{\sqrt {2\pi {\frac {\sigma ^{2}}{n_{0}}}}}}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\&\propto (\sigma ^{2})^{-1/2}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\\&={\frac {(\sigma _{0}^{2}\nu _{0}/2)^{\nu _{0}/2}}{\Gamma (\nu _{0}/2)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+\nu _{0}/2}}}\\&\propto {(\sigma ^{2})^{-(1+\nu _{0}/2)}}\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right].\end{aligned}}}$$

Therefore, the joint prior is

$${\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2};\mu _{0},n_{0},\nu _{0},\sigma _{0}^{2})&=p(\mu \mid \sigma ^{2};\mu _{0},n_{0})\,p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})\\&\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right].\end{aligned}}}$$

The likelihood function from the section above with known variance is:

$${\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\right]\end{aligned}}}$$

Writing it in terms of variance rather than precision, we get: $${\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&\propto {\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}}$$where ${\textstyle S=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.}$

Therefore, the posterior is (dropping the hyperparameters as conditioning factors): $${\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2}\mid \mathbf {X} )&\propto p(\mu ,\sigma ^{2})\,p(\mathbf {X} \mid \mu ,\sigma ^{2})\\&\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right]{\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\\&=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+n_{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}+(n_{0}+n)\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right)\right]\\&\propto (\sigma ^{2})^{-1/2}\exp \left[-{\frac {n_{0}+n}{2\sigma ^{2}}}\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right]\\&\quad \times (\sigma ^{2})^{-(\nu _{0}/2+n/2+1)}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right]\\&={\mathcal {N}}_{\mu \mid \sigma ^{2}}\left({\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}},{\frac {\sigma ^{2}}{n_{0}+n}}\right)\cdot {\rm {IG}}_{\sigma ^{2}}\left({\frac {1}{2}}(\nu _{0}+n),{\frac {1}{2}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right).\end{aligned}}}$$

In other words, the posterior distribution has the form of a product of a normal distribution over ${\textstyle p(\mu |\sigma ^{2})}$ times an inverse gamma distribution over ${\textstyle p(\sigma ^{2})}$, with parameters that are the same as the update equations above.

Occurrence and applications

The occurrence of normal distribution in practical problems can be loosely classified into four categories:

1. Exactly normal distributions;
2. Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
3. Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
4. Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

The ground state of a quantum harmonic oscillator has the Gaussian distribution.

Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:

• Probability density function of a ground state in a quantum harmonic oscillator.
• The position of a particle that experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation  ${\textstyle {\frac {\partial }{\partial t}}f(x,t)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}f(x,t)}$. If the initial location is given by a certain density function ${\textstyle g(x)}$, then the density at time t is the convolution of g and the normal probability density function.

Approximate normality

Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting additively and independently, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.

• In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as
  • Binomial random variables, associated with binary response variables;
  • Poisson random variables, associated with rare events;
• Thermal radiation has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

Histogram of sepal widths for Iris versicolor from Fisher's Iris flower data set, with superimposed best-fitting normal distribution

I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.

— Pearson (1901)

There are statistical methods to empirically test that assumption; see the above Normality tests section.

• In biology, the logarithm of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution (after separation on male/female subpopulations), with examples including:
  • Measures of size of living tissue (length, height, skin area, weight);^[53]
  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
  • Certain physiological measurements, such as blood pressure of adult humans.
• In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb in his works.
• Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.^[54]
• In standardized testing, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

Fitted cumulative normal distribution to October rainfalls, see distribution fitting

• Many scores are derived from the normal distribution, including percentile ranks (percentiles or quantiles), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
• In hydrology the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem.^[55] The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.^[56]

Computational methods

Generating values from normal distribution

The bean machine, a device invented by Francis Galton, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a N(μ, σ²) can be generated as X = μ + σZ, where Z is standard normal. All these algorithms rely on the availability of a random number generator U capable of producing uniform random variates.

• The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ⁻¹(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in Hart (1968) and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places,^[57] which is used by R to compute random variates of the normal distribution.
• An easy-to-program approximate approach that relies on the central limit theorem is as follows: generate 12 uniform U(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6).^[58] Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
• The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1). Then the two random variables X and Y $${\displaystyle X={\sqrt {-2\ln U}}\,\cos(2\pi V),\qquad Y={\sqrt {-2\ln U}}\,\sin(2\pi V).}$$ will both have the standard normal distribution, and will be independent. This formulation arises because for a bivariate normal random vector (X, Y) the squared norm X² + Y² will have the chi-squared distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.
• The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, U and V are drawn from the uniform (−1,1) distribution, and then S = U² + V² is computed. If S is greater or equal to 1, then the method starts over, otherwise the two quantities $${\displaystyle X=U{\sqrt {\frac {-2\ln S}{S}}},\qquad Y=V{\sqrt {\frac {-2\ln S}{S}}}}$$ are returned. Again, X and Y are independent, standard normal random variables.
• The Ratio method^[59] is a rejection method. The algorithm proceeds as follows:
  • Generate two independent uniform deviates U and V;
  • Compute X = √8/e (V − 0.5)/U;
  • Optional: if X² ≤ 5 − 4e^1/4U then accept X and terminate algorithm;
  • Optional: if X² ≥ 4e^−1.35/U + 1.4 then reject X and start over from step 1;
  • If X² ≤ −4 lnU then accept X, otherwise start over the algorithm.

  The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved^[60] so that the logarithm is rarely evaluated.

• The ziggurat algorithm^[61] is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
• Integer arithmetic can be used to sample from the standard normal distribution.^[62] This method is exact in the sense that it satisfies the conditions of ideal approximation;^[63] i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
• There is also some investigation^[64] into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function is widely used in scientific and statistical computing.

The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Different approximations are used depending on the desired level of accuracy.

• Zelen & Severo (1964) give the approximation for Φ(x) for x > 0 with the absolute error |ε(x)| < 7.5·10⁻⁸ (algorithm 26.2.17): $${\displaystyle \Phi (x)=1-\varphi (x)\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\right)+\varepsilon (x),\qquad t={\frac {1}{1+b_{0}x}},}$$ where ϕ(x) is the standard normal probability density function, and b₀ = 0.2316419, b₁ = 0.319381530, b₂ = −0.356563782, b₃ = 1.781477937, b₄ = −1.821255978, b₅ = 1.330274429.
• Hart (1968) lists some dozens of approximations – by means of rational functions, with or without exponentials – for the erfc() function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by West (2009) combines Hart's algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
• Cody (1969) after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
• Marsaglia (2004) suggested a simple algorithm^{[note 1]} based on the Taylor series expansion $${\displaystyle \Phi (x)={\frac {1}{2}}+\varphi (x)\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+{\frac {x^{7}}{3\cdot 5\cdot 7}}+{\frac {x^{9}}{3\cdot 5\cdot 7\cdot 9}}+\cdots \right)}$$ for calculating Φ(x) with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when x = 10).
• The GNU Scientific Library calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomials.
• Dia (2023) proposes the following approximation of ${\textstyle 1-\Phi }$ with a maximum relative error less than ${\textstyle 2^{-53}}$ ${\textstyle \left(\approx 1.1\times 10^{-16}\right)}$ in absolute value: for ${\textstyle x\geq 0}$ ${\textstyle {\begin{aligned}1-\Phi \left(x\right)&=\left({\frac {0.39894228040143268}{x+2.92678600515804815}}\right)\left({\frac {x^{2}+8.42742300458043240x+18.38871225773938487}{x^{2}+5.81582518933527391x+8.97280659046817350}}\right)\\&\left({\frac {x^{2}+7.30756258553673541x+18.25323235347346525}{x^{2}+5.70347935898051437x+10.27157061171363079}}\right)\left({\frac {x^{2}+5.66479518878470765x+18.61193318971775795}{x^{2}+5.51862483025707963x+12.72323261907760928}}\right)\\&\left({\frac {x^{2}+4.91396098895240075x+24.14804072812762821}{x^{2}+5.26184239579604207x+16.88639562007936908}}\right)\left({\frac {x^{2}+3.83362947800146179x+11.61511226260603247}{x^{2}+4.92081346632882033x+24.12333774572479110}}\right)e^{-{\frac {x^{2}}{2}}}\end{aligned}}}$ and for ${\textstyle x<0}$,

$${\displaystyle 1-\Phi \left(x\right)=1-\left(1-\Phi \left(-x\right)\right)}$$

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting p = Φ(z), the simplest approximation for the quantile function is: $${\displaystyle z=\Phi ^{-1}(p)=5.5556\left[1-\left({\frac {1-p}{p}}\right)^{0.1186}\right],\qquad p\geq 1/2}$$

This approximation delivers for z a maximum absolute error of 0.026 (for 0.5 ≤ p ≤ 0.9999, corresponding to 0 ≤ z ≤ 3.719). For p < 1/2 replace p by 1 − p and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation: $${\displaystyle z=-0.4115\left\{{\frac {1-p}{p}}+\log \left[{\frac {1-p}{p}}\right]-1\right\},\qquad p\geq 1/2}$$

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by $${\displaystyle {\begin{aligned}L(z)&=\int _{z}^{\infty }(u-z)\varphi (u)\,du=\int _{z}^{\infty }[1-\Phi (u)]\,du\\[5pt]L(z)&\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&p<1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&p\geq 1/2.\end{cases}}\\[5pt]{\text{or, equivalently,}}\\L(z)&\approx {\begin{cases}0.4115\left\{1-\log \left[{\frac {p}{1-p}}\right]\right\},&p<1/2,\\\\0.4115{\dfrac {1-p}{p}},&p\geq 1/2.\end{cases}}\end{aligned}}}$$

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small relative error on the whole domain for the cumulative distribution function ${\textstyle \Phi }$ and the quantile function ${\textstyle \Phi ^{-1}}$ as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors^[65][66] attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738^{[note 2]} published in the second edition of his The Doctrine of Chances the study of the coefficients in the binomial expansion of (a + b)ⁿ. De Moivre proved that the middle term in this expansion has the approximate magnitude of ${\textstyle 2^{n}/{\sqrt {2\pi n}}}$, and that "If m or ⁠1/2⁠n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ℓ, has to the middle Term, is ${\textstyle -{\frac {2\ell \ell }{n}}}$."^[67] Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.^[68]

Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.

In 1823 Gauss published his monograph "Theoria combinationis observationum erroribus minimis obnoxiae" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M′, M′′, ... to denote the measurements of some unknown quantity V, and sought the most probable estimator of that quantity: the one that maximizes the probability φ(M − V) · φ(M′ − V) · φ(M′′ − V) · ... of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.^{[note 3]} Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:^[69] $${\displaystyle \varphi {\mathit {\Delta }}={\frac {h}{\surd \pi }}\,e^{-\mathrm {hh} \Delta \Delta },}$$ where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear weighted least squares method.^[70]

Pierre-Simon Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.

Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.^{[note 4]} It was Laplace who first posed the problem of aggregating several observations in 1774,^[71] although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e^−t2 dt = √π in 1782, providing the normalization constant for the normal distribution.^[72] Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.^[73]

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss.^[74] His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.^[75]

In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:^[76] The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is $${\displaystyle \operatorname {N} {\frac {1}{\alpha \;{\sqrt {\pi }}}}\;e^{-{\frac {x^{2}}{\alpha ^{2}}}}\,dx}$$

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual.^[77] However, by the end of the 19th century some authors^{[note 5]} had started using the name normal distribution, where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under certain circumstances."^[78] Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.^[79]

Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'.

— Pearson (1920)

Also, it was Pearson who first wrote the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays: $${\displaystyle df={\frac {1}{\sqrt {2\sigma ^{2}\pi }}}e^{-(x-m)^{2}/(2\sigma ^{2})}\,dx.}$$

The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) Introduction to Mathematical Statistics and A. M. Mood (1950) Introduction to the Theory of Statistics.^[80]

See also

• Mathematics portal

• Bates distribution – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
• Behrens–Fisher problem – the long-standing problem of testing whether two normal samples with different variances have same means;
• Bhattacharyya distance – method used to separate mixtures of normal distributions
• Erdős–Kac theorem – on the occurrence of the normal distribution in number theory
• Full width at half maximum
• Gaussian blur – convolution, which uses the normal distribution as a kernel
• Modified half-normal distribution^[81] with the pdf on ${\textstyle (0,\infty )}$ is given as ${\textstyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}$, where ${\textstyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}$ denotes the Fox–Wright Psi function.
• Normally distributed and uncorrelated does not imply independent
• Ratio normal distribution
• Reciprocal normal distribution
• Standard normal table
• Stein's lemma
• Sub-Gaussian distribution
• Sum of normally distributed random variables
• Tweedie distribution – The normal distribution is a member of the family of Tweedie exponential dispersion models.
• Wrapped normal distribution – the Normal distribution applied to a circular domain
• Z-test – using the normal distribution

Notes

1. For example, this algorithm is given in the article Bc programming language.
2. De Moivre first published his findings in 1733, in a pamphlet Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem Expansi that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example Walker (1985).
3. "It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — Gauss (1809, section 177)
4. "My custom of terming the curve the Gauss–Laplacian or normal curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from Pearson (1905, p. 189)
5. Besides those specifically referenced here, such use is encountered in the works of Peirce, Galton (Galton (1889, chapter V)) and Lexis (Lexis (1878), Rohrbasser & Véron (2003)) c. 1875.^{[citation needed]}

References

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External links

• "Normal distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Normal distribution calculator