Question: Let II be a random proportion between 0 and 1 for example, the proportion of black balls in an urn picked at random from some population of urns. Let S be the number of successes in n Bernoulli trials, which given II = p are independent with probability p, for example, the number of black balls in n draws at random, with replacement from the urn picked at random.
a) Find a formula for E(S) in terms of n and E(II).
b) Find a formula for Vαr(S) in terms of n, E(II), and Vαr(II).
c) For given n and E(II) = p, say, which distribution of II makes Vαr(S) as large as possible? Which as small as possible? prove your answers using your answer to b).