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(c) Suppose X and Y are rv's


(c) Suppose X and Y are rv's. Define the rv Z as Z := X + Y. (i) Suppose X ~ Gaussian(3, 4) and Y ~ Gaussian(5, 9). Assume that X and Y are jointly Gaussian with correlation coefficient px,y = 0.5. Then Z ~ Gaussian(uz, oz) is also Gaussian. Determine /z and oz. (ii) Suppose X and Y are independent with X ~ Binomial(10, 0.3) and Y ~ Binomial(20, 0.3). Is Z also binomially distributed? If yes, give its parameters. If no, explain why not.

5. (Marginally Gaussian but jointly non-Gaussian RVs) [10] Consider two jointly distributed RVs X and Y. If they are jointly Gaussian, by definition we know that X and Y are marginally Gaussian as well. However, the converse is not true. Here is a counter example. Let X ~ N(0, 1) and Y = ZX where Z ~ Unif({+1}), Z IL X. a) Show that Y ~ N(0, 1). b) Show that X + Y is not Gaussian. Hence, X and Y are not jointly Gaussian. c) Show that X and Y are uncorrelated but not independent. [4

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