베타분포

2.6.6 베타분포

$B(a,b)$로 표기되는 베타함수는 $a>0,\ b>0$에서

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1) 확률밀도함수

$X\sim \text{BETA}(a,\ b)$로 표기.

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2) 기댓값과 분산

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$\therefore E(X)=\dfrac{\Gamma(a+1)\Gamma(a+b)}{\Gamma(a)\Gamma(a+b+1)}=\dfrac{a}{a+b}\\[10pt]$
$\begin{aligned}\therefore \text{Var}(X)&=\dfrac{\Gamma(a+2)\Gamma(a+b)}{\Gamma(a)\Gamma(a+b+2)}-\left(\dfrac{a}{a+b}\right)^2 \\[10pt] &=\dfrac{(a+1)a}{(a+b+1)(a+b)}-\left(\dfrac{a}{a+b}\right)^2\\[10pt] &=\dfrac{ab}{(a+b+1)(a+b)^2}\end{aligned}$