Q3) Zero-Inflated Poisson, Law of Total Variance Z has the following "zero-inflated Poisson" distribution. Z = XY where Y ~ Pois() = 3) and fx (x) = 0.5 when r E {0, 1}, and fx(x) = 0 otherwise. In short, X has a Bernoulli distribution with p = 0.5, it's a binomial distribution with n = 1, p = 0.5. Find the following: A) P(Z = 0) B) E[Z] C) Var(Z) D) The PDF of Z, fz(z) 3Q4) Negative Binomial Let X1,X2, . . . X,I 2 Geom(p = 1/4) That is, let Xi be identical and independent geometric distributions counting the number of trials until a success (X,: E {1, 2, . . .}), with the success chance of 1/4. Find the following: A) E[X1 + X2] B) Var[X1 + X2] C) P(X1 + X; = 3) Hint: What are all the different ways this can happen? D) P(X1+X2 =4) E) The PMF of Y = X1 + X2, where y 6 {2,3,. . ..}