# Expectation of Beta Distribution

## Theorem

Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ denotes the beta distribution.

Then:

$\expect X = \dfrac \alpha {\alpha + \beta}$

## Proof 1

From the definition of the beta distribution, $X$ has probability density function:

$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$

From the definition of the expected value of a continuous random variable:

$\displaystyle \expect X = \int_0^1 x \, \map {f_X} x \rd x$

So:

 $\ds \expect X$ $=$ $\ds \frac 1 {\map \Beta {\alpha, \beta} } \int_0^1 x^\alpha \paren {1 - x}^{\beta - 1} \rd x$ $\ds$ $=$ $\ds \frac {\map \Beta {\alpha + 1, \beta} } {\map \Beta {\alpha, \beta} }$ Definition 1 of Beta Function $\ds$ $=$ $\ds \frac {\map \Gamma {\alpha + 1} \, \map \Gamma \beta} {\map \Gamma {\alpha + \beta + 1} } \cdot \frac {\map \Gamma {\alpha + \beta} } {\map \Gamma \alpha \, \map \Gamma \beta}$ Definition 3 of Beta Function $\ds$ $=$ $\ds \frac \alpha {\alpha + \beta} \cdot \frac {\map \Gamma \alpha \, \map \Gamma \beta \, \map \Gamma {\alpha + \beta} } {\map \Gamma \alpha \, \map \Gamma \beta \, \map \Gamma {\alpha + \beta} }$ Gamma Difference Equation $\ds$ $=$ $\ds \frac \alpha {\alpha + \beta}$

$\blacksquare$

## Proof 2

 $\ds \expect X$ $=$ $\ds \prod_{r \mathop = 0}^0 \frac {\alpha + r} {\alpha + \beta + r}$ Raw Moment of Beta Distribution $\ds$ $=$ $\ds \frac {\alpha + 0} {\alpha + \beta + 0}$ $\ds$ $=$ $\ds \frac \alpha {\alpha + \beta}$

$\blacksquare$