Mean and Variance of the Beta Distribution
Formula
$X \sim \text{Beta}(\alpha,\beta)$ Surface $$ E(X)={\alpha \over {\alpha + \beta} } \\ \operatorname{Var} (X)={ { \alpha \beta } \over {(\alpha + \beta + 1) { ( \alpha + \beta ) }^2 } } $$
Derivation
Strategy: Direct deduction using the definition of the beta distribution and the basic properties of the gamma function.
Definition of the Beta Distribution: $\alpha , \beta > 0$ A continuous probability distribution $\text{Beta}(\alpha,\beta)$ with the following probability density function is called the Beta distribution. $$ f(x) = { \Gamma (\alpha + \beta) \over { \Gamma (\alpha) \Gamma (\beta) } } x^{\alpha - 1} (1-x)^{\beta - 1} \qquad , x \in [0,1] $$
Recursive formula of the Gamma function: $$ \Gamma (p+1)=p\Gamma (p) $$
Mean
$$ \begin{align*} E(X) =& \int _{0} ^{1} x { \Gamma (\alpha + \beta) \over { \Gamma (\alpha) \Gamma (\beta) } } x^{\alpha - 1} (1-x)^{\beta - 1} dx \\ =& \int _{0} ^{1} { \alpha \over {\alpha + \beta} } { \Gamma (\alpha + \beta + 1) \over { \Gamma (\alpha + 1) \Gamma (\beta) } } x^{\alpha} (1-x)^{\beta - 1} \\ =& { \alpha \over {\alpha + \beta} } \end{align*} $$
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Variance
$$ \begin{align*} E(X^2) =& \int _{0} ^{1} x^2 { \Gamma (\alpha + \beta) \over { \Gamma (\alpha) \Gamma (\beta) } } x^{\alpha - 1} (1-x)^{\beta - 1} dx \\ =& \int _{0} ^{1} { { \alpha ( \alpha + 1 ) } \over { ( \alpha + \beta) ( \alpha + \beta + 1 ) } } { \Gamma (\alpha + \beta + 2) \over { \Gamma (\alpha + 2) \Gamma (\beta) } } x^{\alpha + 1 } (1-x)^{\beta - 1} \\ =& { { \alpha ( \alpha + 1 ) } \over { ( \alpha + \beta) ( \alpha + \beta + 1 ) } } \end{align*} $$ Therefore, $$ \begin{align*} \operatorname{Var} (X) =& { { \alpha ( \alpha + 1 ) } \over { ( \alpha + \beta) ( \alpha + \beta + 1 ) } } - { { \alpha \alpha } \over {(\alpha + \beta)(\alpha + \beta)} } \\ =& { \alpha \over {\alpha + \beta} } { { (\alpha + 1) (\alpha + \beta) - \alpha (\alpha + \beta + 1) } \over { (\alpha + \beta) (\alpha + \beta + 1) } } \\ =& { { \alpha \beta } \over {(\alpha + \beta + 1) { ( \alpha + \beta ) }^2 } } \end{align*} $$
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