33. Let X and Y be independent exponential random variables with respective rates and . (a)...

33. Let X and Y be independent exponential random variables with respective rates λ and μ.

(a) Argue that, conditional on X>Y , the random variables min(X,Y) and X − Y are independent.

(b) Use part

(a) to conclude that for any positive constant c E[min(X,Y)|X>Y + c] = E[min(X,Y)|X>Y]

= E[min(X,Y)] =

1

λ + μ

(c) Give a verbal explanation of why min(X,Y) and X − Y are (unconditionally) independent.