# Consider generating a random variable Y by the following mechanism.

**Question description**

Consider generating a random variable Y by the following mechanism. First, generate X = x distributed as a Gamma(α, λ) random variable, then generate Y as an exponential with rate parameter x.

- (a) Find P(Y<1).
- (b) Find the conditional distribution of X given Y = y.
- (c) Find the conditional expectation and variance of X given Y = y (if you carefully examine your answer for part (b), you won’t need to do any integrals).
- (d) Are X and Y independent random variables? Give both an intuitive and mathematical justification for your reasoning.

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