Chapter 5.2, Problem 8E

Let X have a beta distribution with parameters $\alpha$ and $\beta$.

(See Example 5.2-3.)

(a) Show that the mean and variance of X are, respectively.

$\mu =\frac{\alpha }{\alpha +\beta }$ and ${\sigma }^2=\frac{\alpha \beta }{\left(\alpha +\beta +1\right){\left(\alpha +\beta \right)}^2}$

(b) Show that when $\alpha >1$ and $\beta >1$, the mode is at $x=\frac{\left(\alpha -1\right)}{\left(\alpha +\beta -2\right)}$.