Question

Let X and Y again be uniformly distributed independent random variables on [0, 1]. a) Compute the expected value E(XY ). b) What is the probability density function fZ(z) of Z = XY ? Hint: First compute the cumulative distribution function FZ(z) = P(Z ≤ z) using a double integral, and then differentiate in z. c) Use your answer to b) to compute E(Z). Compare it with your answer to a)

Answer 1

a) Computing the expected value E(XY) gives 1/4

b) The probability density function fZ(z) of Z = XY is calculated in the attached picture.

c) Computing E(Z) gives 1/4

Explanation:

Comparing the computation of E(Z) using the answer to b), it shows that the values are equal.