Let X₁, X2, X3, ..., Xn be a random sample of size n from population X. Suppose that X~N (8,1) Σ and Y = 1X₁-0√n. √n ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y2 is between 0.02 and 5.02?

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A: Please see the handwritten solution.