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.

P(lnH≥z)=

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

f(z)=