(a) For the hierarchy in Example 4.4.6, show that the variance of X can be written
Var X = nEP(l - BP) + n(n - 1) Var P.
(The first term reflects binomial variation with success probability EP, and the second term is often called "extra-binomial" variation, showing how the hierarchical model has a variance that is larger than the binomial alone.)€ƒ
(b) For the hierarchy in Exercise 4.32, show that the variance of Y can be written

where μ = Eˆ§. Identify the "extra-Poisson" variation induced by the hierarchy.

Var Y = EA + Var A = u+