Let X1, . . . , Xn be an independent and identically distributed sequence of Exponential(1) random variables, where n 3. Find the conditional probability density function for the maximum M=X(n) given that the second smallest X(2)=1 . Use this to verify that, in the case that n=4 , E(MX(2)=1)=e2/22e .

Hint: Start by using the "PDF method" to find the density for X(2) and the joint density for (M,X(2)) .