29. Let X and Y be independent exponential random variables with respective rates and , where...

29. Let X and Y be independent exponential random variables with respective rates λ and μ, where λ>μ. Let c > 0.

(a) Show that the conditional density function of X, given that X + Y =

c, is fX|X+Y (x|c) = (λ − μ)e−(λ−μ)x 1 − e−(λ−μ)c , 0

(b) Use part

(a) to find E[X|X + Y = c].

(c) Find E[Y|X + Y = c].