Question

Let U1 and U2 be independent Uniform(0, 1) random variables and let Y = U1U2.

(a) Write down the joint pdf of U1 and U2.

(b) Find the cdf of Y by obtaining an expression for FY (y) = P(Y ≤ y) = P(U1U2 ≤ y) for all y.

(c) Find the pdf of Y by taking the derivative of FY (y) with respect to y

(d) Let X = U2 and find the joint pdf of the rv pair (X, Y ) using the bivariate transformation method. Define the joint support of (X,Y).

(e) Integrate the joint pdf of (X, Y ) over X in order to get the pdf of Y.

Answer A, D, E.

Answer 1

(a) Joint PDF of U1 and U2 is given by

Since U1 and U2 are independent so joint PDF is the product of marginal pdfs

(D) Given