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Mean and Variance of the Beta Distribution 📂Probability Distribution

Mean and Variance of the Beta Distribution

Formula

X∼Beta(α,β)X \sim \text{Beta}(\alpha,\beta) Surface E(X)=αα+βVar⁡(X)=αβ(α+β+1)(α+β)2 E(X)={\alpha \over {\alpha + \beta} } \\ \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: α,β>0\alpha , \beta > 0 A continuous probability distribution Beta(α,β)\text{Beta}(\alpha,\beta) with the following probability density function is called the Beta distribution. f(x)=Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1,x∈[0,1] 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: Γ(p+1)=pΓ(p) \Gamma (p+1)=p\Gamma (p)

Mean

E(X)=∫01xΓ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1dx=∫01αα+βΓ(α+β+1)Γ(α+1)Γ(β)xα(1−x)β−1=αα+β \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

E(X2)=∫01x2Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1dx=∫01α(α+1)(α+β)(α+β+1)Γ(α+β+2)Γ(α+2)Γ(β)xα+1(1−x)β−1=α(α+1)(α+β)(α+β+1) \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, Var⁡(X)=α(α+1)(α+β)(α+β+1)−αα(α+β)(α+β)=αα+β(α+1)(α+β)−α(α+β+1)(α+β)(α+β+1)=αβ(α+β+1)(α+β)2 \begin{align*} \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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