Exercise 20.8 [beta-integration-exercise]

This exercise investigates properties of the Beta distribution defined in Equation (beta-equation).

1. By integrating over the range $[0,1]$, show that the normalization constant for the distribution $[a,b]$ is given by $\alpha = \Gamma(a+b)/\Gamma(a)\Gamma(b)$ where $\Gamma(x)$ is the Gamma function, defined by $\Gamma(x+1)x\cdot\Gamma(x)$ and $\Gamma(1)1$. (For integer $x$, $\Gamma(x+1)x!$.)

2. Show that the mean is $a/(a+b)$.

3. Find the mode(s) (the most likely value(s) of $\theta$).

4. Describe the distribution $[\epsilon,\epsilon]$ for very small $\epsilon$. What happens as such a distribution is updated?