Suppose that x and y are independent exponential random

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Consider the experiments: First draw a Poisson random variable N with parameter ?. Then perform N independent Bernoulli trial each with success probability p. Let Z be the total number of successes in the whole experiment. What are the mean and variance of Z?

Suppose that X and Y are independent exponential random variables each with rate ?. What is the conditional probability of X, given that Z = X + Y = z?

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