(a) Let X1,X2 Exponential() be

(a) Let X1,X2 ∼ Exponential(λ) be independent, λ > 0. Calculate the density of Y := X1 + X2. [Hint: One way to approach this problem would be to compute the CDF of Y and then differentiate the CDF.]

(b) Let t > 0. What is the density of X1, conditioned on X1 +X2 = t? [Hint: Once again, it may be helpful to consider the CDF P(X1 ≤ x | X1 + X2 = t). To tackle the conditioning part, try conditioning instead on the event {X1 +X2 ∈ [t,t +ε]}, where ε > 0 is small.]