Let \(Y_{i}\) be the number of successes in \(n_{i}\) trials with
$$Y_{i} \sim \operatorname{Bin}\left(n_{i}, \pi_{i}\right)$$
where the probabilities \(\pi_{i}\) have a Beta distribution
$$\pi_{i} \sim \operatorname{Be}(\alpha, \beta)$$
The probability density function for the beta distribution is \(f(x ; \alpha,
\beta)=\) \(x^{\alpha-1}(1-x)^{(\beta-1)} / B(\alpha, \beta)\) for \(x\) in \([0,1],
\alpha > 0, \beta > 0\) and the beta
function \(B(\alpha, \beta)\) defining the normalizing constant required to
ensure that \(\int_{0}^{1} f(x ; \alpha, \beta) d x=1 .\) It can be shown that
\(\mathrm{E}(X)=\alpha /(\alpha+\beta)\) and \(\operatorname{var}(X)=\)
\(\alpha \beta /\left[(\alpha+\beta)^{2}(\alpha+\beta+1)\right] .\) Let
\(\theta=\alpha /(\alpha+\beta),\) and hence, show that
(a) \(\mathrm{E}\left(\pi_{i}\right)=\theta\)
(b) \(\operatorname{var}\left(\pi_{i}\right)=\theta(1-\theta)
/(\alpha+\beta+1)=\phi \theta(1-\theta)\)
\((\mathrm{c}) \mathrm{E}\left(Y_{i}\right)=n_{i} \theta\)
(d) \(\operatorname{var}\left(Y_{i}\right)=n_{i}
\theta(1-\theta)\left[1+\left(n_{i}-1\right) \phi\right]\) so that
\(\operatorname{var}\left(Y_{i}\right)\) is larger than the
Binomial variance (unless \(n_{i}=1\) or \(\phi=0\) ).
