Question: Let X be a random variable defined as the sum of N independent Bernoulli trials in which the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a random variable that behaves according to a Poisson distribution function with the parameter λ.

1. Derive the conditional distribution function of X given N = n and state your reasoning behind your derivation.
2. Derive the joint distribution function of X and N and state your reasoning behind your derivation.
3. Without explicitly calculating it, what would you expect the correlation coefficient between X and N to be? (i.e. negative? zero? positive?) Mark sure to provide your reasoning.
4. Without explicitly calculating the marginal distribution function of X, briefly describe the effect of the Poisson parameter λ on the unconditional mean of X (i.e. what happens to the unconditional mean of X as we change λ?). Make sure to provide your reasoning.

BONUS: Explicitly derive the marginal distribution function of X

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