(12 marks) Recall that X is a standard normal random variable if its probability density function...

 

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(12 marks) Recall that X is a standard normal random variable if its probability density function is f(x) = 2 ex (a) Compute the moment generating function Mx2(t) := f(x) = 1 without providing a proof. (b) Use (a), or otherwise, to compute E[X4]. E[etX2]. You may use the fact that k i=1 (c) Let X1,..., Xk be independent standard normal random variables, and Y = - 1X2. Derive a Chernoff bound to prove that Pr(Y > 1+ ) e-ke/8 for 0 1. You may use the Taylor expansion ln(1 x) = x/i for 1 x 1. (12 marks) Recall that X is a standard normal random variable if its probability density function is f(x) = 2 ex (a) Compute the moment generating function Mx2(t) := f(x) = 1 without providing a proof. (b) Use (a), or otherwise, to compute E[X4]. E[etX2]. You may use the fact that k i=1 (c) Let X1,..., Xk be independent standard normal random variables, and Y = - 1X2. Derive a Chernoff bound to prove that Pr(Y > 1+ ) e-ke/8 for 0 1. You may use the Taylor expansion ln(1 x) = x/i for 1 x 1.