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

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) .