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

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.