8. (Total: 12 points) (a) Suppose Y ~Po(0) and let Z = Y +1. (i) (2...

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8. (Total: 12 points) (a) Suppose Y ~Po(0) and let Z = Y +1. (i) (2 points) Determine the probability mass function of Z. (ii) (2 points) Determine the mean and variance of Z. (b) Suppose N~ Geo(p) and the conditional distribution of X given N = n is (n, ). (i) (6 points) Find the conditional probability mass function of N given X = x > 0. (ii) (2 points) Using the results of (a), or otherwise, determine E(N|X = x) and Var(N|Xx). 8. (Total: 12 points) (a) Suppose Y ~Po(0) and let Z = Y +1. (i) (2 points) Determine the probability mass function of Z. (ii) (2 points) Determine the mean and variance of Z. (b) Suppose N~ Geo(p) and the conditional distribution of X given N = n is (n, ). (i) (6 points) Find the conditional probability mass function of N given X = x > 0. (ii) (2 points) Using the results of (a), or otherwise, determine E(N|X = x) and Var(N|Xx).