Let Z be a standard normal random variable. a Show that the expected values of all odd

Chapter 4, Problem 4.199

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Let Z be a standard normal random variable. a Show that the expected values of all odd integer powers of Z are 0. That is, if i = 1, 2,... , show that E(Z2i1) = 0. [Hint: A function g() is an odd function if, for all y, g(y) = g(y). For any odd function g(y), # g(y) dy = 0, if the integral exists.] b If i = 1, 2,... , show that E(Z2i ) = 2i i + 1 2 . [Hint: A function h()is an even function if, for all y, h(y) = h(y). For any even function h(y), # h(y) dy = 2 # 0 h(y) dy, if the integrals exist. Use this fact, make the change of variable w = z2/2, and use what you know about the gamma function.] c Use the results in part (b) and in Exercises 4.81(b) and 4.194 to derive E(Z2), E(Z4), E(Z6), and E(Z8). d If i = 1, 2,... , show that E(Z2i ) = 'i j=1 (2 j 1). This implies that the ith even moment is the product of the first i odd integers.