3 Variance of exponential and uniform distributions a Comput

3. Variance of exponential and uniform distributions

(a) Compute Var[Y ] when Y Exp(1) and when Y Unif(0, 1).

(b) Show that

i. if Y Unif(a, b), then X = (Y a)/(b a) Unif(0, 1), where a < b.

ii. if Y Exp(), then X = Y Exp(1) where > 0. Hint: Compute FX(x) = P(X x) = P(Y ?).

(c) Combine (a) and (b) to obtain Var[X] when X Exp() and when X Unif(a, b).

Solution