# Thread: mean and variance of a beta distribution

1. ## mean and variance of a beta distribution

Let $\displaystyle X$ have a beta distribution $\displaystyle f(x)=\frac{\gamma(\alpha+\beta)}{\gamma(\alpha)\ga mma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},0<x<1$. Show that the mean and variance of $\displaystyle X$ are $\displaystyle \mu=\frac{\alpha}{\alpha+\beta}$ and $\displaystyle \sigma^2=\frac{\alpha\beta}{(\alpha+\beta+1)(\alph a+\beta)^2}$.

$\displaystyle \mu=\int_0^1x\frac{\gamma(\alpha+\beta)}{\gamma(\a lpha)\gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}dx=\int_0^1\frac{\gamma(\alpha+\beta)}{\gamma(\a lpha)\gamma(\beta)}x^{\alpha}(1-x)^{\beta-1}dx$ I can't figure out how to integrate this. Can I get some help please?

2. Originally Posted by dori1123
Let $\displaystyle X$ have a beta distribution $\displaystyle f(x)=\frac{\gamma(\alpha+\beta)}{\gamma(\alpha)\ga mma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},0<x<1$. Show that the mean and variance of $\displaystyle X$ are $\displaystyle \mu=\frac{\alpha}{\alpha+\beta}$ and $\displaystyle \sigma^2=\frac{\alpha\beta}{(\alpha+\beta+1)(\alph a+\beta)^2}$.

$\displaystyle \mu=\int_0^1x\frac{\gamma(\alpha+\beta)}{\gamma(\a lpha)\gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}dx=\int_0^1\frac{\gamma(\alpha+\beta)}{\gamma(\a lpha)\gamma(\beta)}x^{\alpha}(1-x)^{\beta-1}dx$ I can't figure out how to integrate this. Can I get some help please?
$\displaystyle \mu=\int_0^1x\frac{\gamma(\alpha+\beta)}{\gamma(\a lpha)\gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}dx$

$\displaystyle =\int_0^1\frac{\gamma(\alpha+\beta)}{\gamma(\alpha )\gamma(\beta)}x^{\color{red}{(\alpha+1)-1}}$ $\displaystyle (1-x)^{\beta-1}dx$

$\displaystyle =\frac{\gamma(\alpha+\beta)}{\gamma(\alpha)\gamma( \beta)}\cdot \frac{1}{\frac{\gamma(\alpha+1+\beta)}{\gamma(\alp ha+1)\gamma(\beta)}}\int_0^1 \underbrace{\frac{\gamma(\alpha+1+\beta)}{\gamma(\ alpha+1)\gamma(\beta)} x^{(\alpha+1)-1}(1-x)^{\beta-1}}dx$ $\displaystyle \color{red}{\text{This is the pdf of a } \beta(\alpha+1,\beta) \text{ r.v.}}$

$\displaystyle =\frac{\gamma(\alpha+\beta)}{\gamma(\alpha)\gamma( \beta)}\cdot \frac{1}{\frac{\gamma(\alpha+1+\beta)}{\gamma(\alp ha+1)\gamma(\beta)}}\cdot 1$

Apply the same technique to obtain the second moment.

aNon1