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1 Expectation, Co-variance and Independence [25pts] Suppose X, Y and Z are three different random variables. Let X obeys Bernouli Distribution. The probability disbribution function is 0.5 x=1 0.5 x=-1 Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y are independent. Meanwhile, let Z = XY. N(0,1). X and Y (a) What is the Expectation (mean value) of X? 3pts (b) Are Y and Z independent? (Just clarify, do not need to prove) [2pts c) Show that Z is also a standard Normal (Gaussian) distribution, which means Z~N(0,1). [10pts (hint: Use the denominator part of the equation 1 on other cases at the page 16 of the slide 3.) (d) Are Y and Z uncorrelated(which means Cor(Y,Z) = 0)? (need to prove) [10pts] (hint: You may need this Theorem about Independence and Functions of Random Variables. Let X and Y be independent randomi variables. Then, U = g(X and V = h(Y) are also independent for any function g and h.)

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