From a set of n elements, a nonempty subset is chosen at random in the sense that

Question:

From a set of n elements, a nonempty subset is chosen at random in the sense that all of the nonempty subsets are equally likely to be selected. Let X denote the number of elements in the chosen subset. Using the identities given in Theoretical Exercise 12 of Chapter 1, show that

Show also that, for n large,
Var(X) ~ n/4
in the sense that the ratio Var(X) to n/4 approaches 1 as n approaches ˆž. Compare this formula with the limiting form of Var(Y) when
P{Y = i} = 1/n, i = 1, . . . , n.