Let X and Y be two independent geometric random variables wh

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)].

Solution

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

we know that E[max(X,Y)]= E[X] + E[Y] - E[min(X,Y)]

but as per question we want only E[min(X,Y)]

The value of min(X,Y) is a geometric random variable mean p+q-pq.

Therefore E[min(X,Y)] = 1 / (p+q-pq).