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.
(a) Derive the conditional distribution function of X given N = n and state your reasoning behind your derivation.
(b) Derive the joint distribution function of X and N and state your reasoning behind your derivation.
(c) Without explicitly calculating it, what would you expect the correlation coecient between X and N to be? (i.e. negative? zero? positive?) Mark sure to provide your reasoning.
(d) Without explicitly calculating the marginal distribution function of X, briefly describe the e↵ect of the Poisson parameter on the mean of X (i.e. what happens to the mean of X as we change ?). Make sure to provide your reasoning.
BONUS: Explicitly derive the marginal distribution function of X