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.

1. 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