1. Let X1, X2, . . . be independent random variables, each with mean and variance 2, and let N be a random

1. Let X1, X2, . . . be independent random variables, each with mean μ and variance σ 2, and let N be a random variable which takes values in the positive integers {1, 2, . . . } and which is independent of the Xi . Show that the sum S = X1 + X2 + · · · + XN has variance given by var(S) = σ 2E(N) + μ2 var(N).

If Z0, Z1, . . . are the generation sizes of a branching process in which each family size has mean μ and variance σ 2, use the above fact to show that var(Zn) = σ 2μn−1 + μ2 var(Zn−1),

= μ var(Zn−1) + σ 2μ2n−2.