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EE 322 Probabilistic Methods for Electrical Engineers Homework 7 Assigned: 03/15. 03/27 Problem 1. Let X be a random variable such that M (s) = a +...

i need you to do problem 2 of the attached file.
EE 322 Probabilistic Methods for Electrical Engineers — Homework 7 Assigned: 03/15. Due: 03/27 Problem 1. Let X be a random variable such that M ( s ) = a + be 2 s + ce 4 s , E[ X ] = 3 , Var( X ) = 2 . Find a , b , and c , and the PMF of X . Problem 2. Let X and Y be two independent standard Gaussian random variables. De±ne Z = X · Y . We would like to show that Z is not Gaussian distributed. First, ±nd the MGF of Z . You may use any result that we have derived or you can ±nd. Second, argue that Z is not Gaussian distributed based on the MGF of it. Problem 3. Let X 1 , X 2 , . . . , X 50 be ±fty i.i.d. (independent and identically distributed) random variables. They all have the common PDF f X i ( x ) = e - x u ( x ) (1) where u ( x ) is the step function. De±ne Y = 50 i =1 X i . (a) Find the mean and variance of Y (b) Find the MGF of Y . (c) Find the exact distribution of Y (d) Find the approximate distribution of Y using Central Limit Theorem (that is, assuming Y is Gaussian with its mean and variance.) (e) Compare the two distributions by plotting them in the same ±gure. END OF ASSIGNMENT 1
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