Question: 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(λ) by making small, but that also constrains the mean and variance of X to be small since both are ). The Zero-Inflated Poisson distribution is a modification of the Poisson to address this issue, making it easier to handle frequent zero values gracefully. A Zero-Inflated Poisson r.v. X with parameters p and can be generated as follows. First flip 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(λ). 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-Inflated Poisson r.v. X.
(b) Explain why X has the same distribution as (1 I)Y , where I Bern(p) is independent of Y Pois(λ).
(c) Find the mean of X in two different ways: directly using the PMF of X, and using the representation from (b). For the latter, 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 X.