Question

1) Let X and Y be independent continuous random variables that are uniformly distributed on (0,1) . Let H=(X+2)Y . Find the probability P(lnH≥z) where z is a given number that satisfies e^z<2 . Your answer should be a function of z .

Hint: Condition on X .

2) Let X be a standard normal random variable, and let FX(x) be its CDF. Consider the random variable Z=FX(X) . Find the PDF fZ(z) of Z . Note that fZ(z) takes values in (0,1) .