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4. (a) For the hierarchical model Y|A Poisson(A) and A Gamma(a,b), find the marginal distribution, mean, and variance of Y. Show that the marginal distribution of Y is a negative binomial if a is an integer. (b) Show that the three-stage model YA ~ Binomial(N,P), NA ~ Poisson(A), and A ~ Gammala, B) leads to the same marginal distribution of Y. 2. Let X and Y are independent Poisson random variables with parameter and pe respectively. (a) Show that X + Y is a Poisson random variable with parameter +H. (b) What is the conditional probability P(X = X(X + Y = n). Show your steps. (c) Hence, are X and X+Y independent?

Problem 11 (1pt) You are in an apple orchard and looking to pick some juicy Honeycrisps. The orchard only has Honeycrisps and Macintosh apples. The probability of picking a Honeycrisp is 0.3 throughout the entire orchard and each apple you pick is independent of all other apple (these are special apples trees where the apples are independent of each other). You want to model the number of Honeycrisps in a batch of 64 apples. 11. Which distribution should you use? a. Geometric b. Binomial C. Negative Binomial d. Hypergeometric e. Discrete Poisson f. Normal g. Standard Normal h. Gamma i. Exponential j. Chi-Square