Solve these macroeconomic problems.

Two actuaries are interested in combining Poisson distributions. John states that the difference between two independent Poisson distributions has a Poisson distribution, ie if X ~ Poi(1) and X, ~ Poi(/) then X - X, ~ Poi(1-U). Jennie states that the sum of two independent Poisson distributions has a Poisson distribution, ie if X ~ Poi(1) and X, ~ Poi(/) then X, + X, ~ Poi(1 + !!). Use moment generating functions to examine John and Jennie's claims. [5]Question 2.18 Let X be a random variable with mean 3 and standard deviation 2, and let Y be a random variable with mean 4 and standard deviation 1. X and Y have a correlation coefficient of -0.3. Let Z = X +Y . Calculate: (i) cov(X, Z) [2] (ii) var(Z) . [2] [Total 4] Question 2.19 X has a Poisson distribution with mean 5 and Y has a Poisson distribution with mean 10. If cov(X, Y) =-12, calculate the variance of Z where Z = X -2Y +3. [2]A random sample of n observations is taken from a normal distribution with variance o'. The sample variance is an observation of a random variable S. Show that: (i) E(S2) =02 [2] (ii) var($2) - 204 1-1 (2] [Total 4]Given the following equations, derive the LM curve equation. L = Y-100r (Real mone3y demand) m = 295 (real money supply)