Let X and Y be two independent geometric random variables, where X has parameter p and Y has parameter q.

Find E [min(X, Y )] .

Hint:

A similar problem to compute E [max(X, Y )] is discussed in the lecture notes. When a random variable Z takes on non-negative integer values then its expectation equals the following sum: E [Z] = X i1 Pr(Z i) = X i0 Pr(Z > i) . We will use that formula for E [Z] when Z = min(X, Y )

We need to leverage the independence of X and Y . Here is what we can do: Pr(Z > i) = Pr(X > i and Y > i) = Pr(X > i) Pr(Y > i) What is Pr(X > i)? It is (1 p) i because this means that i trials resulted in failures. What is Pr(Y > i)? It is (1 q) i because this means that i trials resulted in failures.