# probability

Problem 2. Continuous Random Variables
2 points possible (graded, results hidden)
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 𝑒𝑧<2 . Your answer should be a function of 𝑧 .

Hint: Condition on 𝑋 .

1. P(ln H > z) = unanswered

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) .

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