2.1 Referring to Example 1.5, suppose that X has the binomial distribution b(p, n) and the family of prior distributions for p is the family of beta distributions B

(a, b).

(a) Show that the marginal distribution of X is the beta-binomial distribution with mass function

n x

 H(a + b)

H(a)H(b)

H(x + a)H(n − x + b)

H(n + a +

b) .

(b) Show that the mean and variance of the beta-binomial is given by EX = na a + b and var X = n a

a + b

 b a + b

 a + b + n a + b + 1

.

[Hint: For part (b), the identities EX = E[E(X|p)] and var X = var[E(X|p)] +

E[var(X|p)] are helpful.]