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

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

Show also that for n large,

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

Exercise

Consider the following combinatorial identity:

(a) Present a combinatorial argument for this identity by considering a set of n people and determining, in two ways, the number of possible selections of a committee of any size and a chairperson for the committee.

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(b) Verify the following identity for n = 1, 2, 3, 4, 5:

For a combinatorial proof of the preceding, consider a set of n people and argue that both sides of the identity represent the number of different selections of a committee, its chairperson, and its secretary (possibly the same as the chairperson).

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(c) Now argue that