(Solution) - a For the hierarchy in Example 4 4 6 show that the -(2025 Original AI-Free Solution)

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(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.)?
(b) For the hierarchy in Exercise 4.32, show that the variance of Y can be written