Distribución beta

Si una variable aleatoria continua ${\displaystyle X}$ tiene una distribución beta con parámetros ${\displaystyle \alpha ,\beta >0}$ entonces escribiremos ${\displaystyle X\sim \mathrm {B} (\alpha ,\beta )}$.

Otras notaciones para la distribución beta usadas son ${\displaystyle X\sim \operatorname {Beta} (\alpha ,\beta )}$, ${\displaystyle X\sim {\mathcal {Be}}(\alpha ,\beta )}$ o ${\displaystyle X\sim \beta _{\alpha ,\beta }}$.

La función de densidad de ${\displaystyle X}$ es

${\displaystyle f_{X}(x)={\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}}$

para valores ${\displaystyle 0<x<1}$ donde ${\displaystyle \mathrm {B} (\alpha ,\beta )}$ es la función beta y se define para ${\displaystyle \alpha ,\beta >0}$ como

${\displaystyle \mathrm {B} (\alpha ,\beta )=\int _{0}^{1}x^{\alpha -1}(1-x)^{\beta -1}dx}$

y algunas de las propiedades que satisface son:

1. ${\displaystyle \mathrm {B} (\alpha ,\beta )=\mathrm {B} (\beta ,\alpha )}$
2. ${\displaystyle \mathrm {B} (\alpha ,\beta )={\frac {\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}}}$

La función de distribución de ${\displaystyle X}$ es

${\displaystyle F_{X}(x)={\frac {\mathrm {B} (x;\alpha ,\beta )}{\mathrm {B} (\alpha ,\beta )}}=I_{x}(\alpha ,\beta )}$

donde ${\displaystyle \mathrm {B} (x;\alpha ,\beta )}$ es la función beta incompleta y ${\displaystyle I_{x}(\alpha ,\beta )}$ es la función beta incompleta regularizada.

La media de la variable aleatoria ${\displaystyle X}$ es

${\displaystyle {\text{E}}[X]={\frac {\alpha }{\alpha +\beta }}}$

La varianza de la variable aleatoria ${\displaystyle X}$ es

${\displaystyle {\text{Var}}(X)={\frac {\alpha \beta }{(\alpha +\beta +1)(\alpha +\beta )^{2}}}}$.

La moda de la variable aleatoria ${\displaystyle X}$ es

${\displaystyle {\frac {\alpha -1}{\alpha +\beta -2}}}$

para valores de ${\displaystyle \alpha ,\beta >1}$.

El ${\displaystyle n}$-ésimo momento de ${\displaystyle X}$ es

${\displaystyle {\begin{aligned}{\text{E}}[X^{n}]&={\frac {\mathrm {B} (\alpha +n,\beta )}{\mathrm {B} (\alpha +\beta )}}\\&=\prod _{r=0}^{n-1}{\frac {\alpha +r}{\alpha +\beta +r}}\\&={\frac {\alpha (\alpha +1)\cdots (\alpha +n-1)}{(\alpha +\beta )(\alpha +\beta +1)\cdots (\alpha +\beta +n-1)}}\end{aligned}}}$

para ${\displaystyle n\in \mathbb {N} }$.

La función generador de momentos de la variable aleatoria ${\displaystyle X}$ está dada por

${\displaystyle {\begin{aligned}M_{X}(t)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}{\frac {\mathrm {B} (\alpha +n,\beta )}{\mathrm {B} (\alpha ,\beta )}}\\&=1+\sum _{n=1}^{\infty }\left(\prod _{r=0}^{n-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{n}}{n!}}\end{aligned}}}$

El logaritmo de la media geométrica ${\displaystyle G_{X}}$ de una distribución con variable aleatoria ${\displaystyle X}$ es la media aritmética de ${\displaystyle \ln(X)}$ o equivalentemente, su valor esperado:

${\displaystyle \ln G_{X}=\operatorname {E} [\ln X]}$

Para una distribución beta:

${\displaystyle {\begin{aligned}\operatorname {E} [\ln X]&=\int _{0}^{1}\ln xf_{X}(x)dx\\&=\int _{0}^{1}\ln x\;{\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}dx\\&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}\int _{0}^{1}{\frac {\partial x^{\alpha -1}(1-x)^{\beta -1}}{\partial \alpha }}\;dx\\&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\frac {\partial }{\partial \alpha }}\int _{0}^{1}x^{\alpha -1}(1-x)^{\beta -1}dx\\&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\frac {\partial \mathrm {B} (\alpha ,\beta )}{\partial \alpha }}\\&={\frac {\partial \ln \mathrm {B} (\alpha ,\beta )}{\partial \alpha }}\\&={\frac {\partial \ln \Gamma (\alpha )}{\partial \alpha }}-{\frac {\partial \ln \Gamma (\alpha +\beta )}{\partial \alpha }}\\&=\psi (\alpha )-\psi (\alpha +\beta )\end{aligned}}}$

donde ${\displaystyle \psi }$ es la función digamma.

• Si ${\displaystyle X\sim \mathrm {B} (\alpha ,\beta )}$ entonces ${\displaystyle 1-X\sim \mathrm {B} (\beta ,\alpha )}$.
• Si ${\displaystyle X\sim \mathrm {B} (\alpha ,\beta )}$ entonces ${\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}$, la distribución beta de segundo orden.
• Si ${\displaystyle X\sim \mathrm {B} \left({\frac {n}{2}},{\frac {m}{2}}\right)}$ entonces ${\displaystyle {\frac {mX}{n(1-X)}}\sim F_{n,m}}$.
• Si ${\displaystyle X\sim \mathrm {B} (\alpha ,1)}$ entonces ${\displaystyle -\ln(X)\sim \operatorname {Exponencial} (\alpha )}$.

• Si ${\displaystyle X\sim \mathrm {B} (1,1)}$ entonces ${\displaystyle X\sim \operatorname {U} (0,1)}$.
• ${\displaystyle \lim _{n\to \infty }n\mathrm {B} (1,n)=\operatorname {Exponencial} (1)}$.
• ${\displaystyle \lim _{n\to \infty }n\mathrm {B} (k,n)=\Gamma (k,1)}$.
• Un caso partícular de la Distribución Beta es la Distribución PERT que toma tres parámetros: Optimista, más frecuente y pesimista.