In many problems about modeling count data, it is found that values of zero in the data are far more common than can be explained well using a Poisson model (we can make P(X =0) large for X Pois() by making small, but that also constrains the mean and variance of X to be small since both are ). The Zero-Inated Poisson distribution is a modication of the Poisson to address this issue, making it easier to handle frequent zero values gracefully.

A Zero-Inated Poisson r.v. X with parameters p and can be generated as follows. First ip a coin with probability of p of Heads. Given that the coin lands Heads, X = 0. Given that the coin lands Tails, X is distributed Pois(). Note that if X = 0 occurs, there are two possible explanations: the coin could have landed Heads (in which case the zero is called a structural zero), or the coin could have landed Tails but the Poisson r.v. turned out to be zero anyway. For example, if X is the number of chicken sandwiches consumed by a random person in a week, then X =0 for vegetarians (this is a structural zero), but a chicken-eater could still have X =0 occur by chance (since they might not happen to eat any chicken sandwiches that week).

(a) Find the PMF of a Zero-Inated Poisson r.v. X.

(b) Explain why X has the same distribution as (1I)Y, where I Bern(p) is independent of Y Pois().

(c) Find the mean of X using the representation from (b). You can use the fact that if r.v.s Z and W are independent, then E(ZW)= E(Z)E(W).

(d) Find the variance of X.