Question: Let X1, . . . , Xn be i.i.d. random variables with PMF PX(x) = 1/2 x = 2 1/2 x = 4 0 otherwise.

Let X1, . . . , Xn be i.i.d. random variables with PMF PX(x) = 1/2 x = 2 1/2 x = 4 0 otherwise. Let S300 = X1 + + X300. (a) Determine the mean of and variance of S300. (b) Use the Central Limit Theorem approximation to estimate the probability P S300 E[S300] 40 . You can leave your answer in terms of the standard normal CDF (z). (c) Suppose you did not know the PMF PX(x), but you knew that the standard deviation of the random variables Xk was one. You measure X1, . . . , X300 and compute the sample mean M300 = 1 300 P300 k=1 Xk. Find a symmetric confidence interval for the true mean around the observed value M300 with confidence level 0.95. Use the following assumptions: Q(1.28) = 1 (1.28) = 0.1; Q(1.645) = 1 (1.645) = 0.05; Q(1.96) = 1 (1.96) = 0.025. (d) Suppose you also did not know the standard deviation, but you were able to compute the sample variance of the 300 samples Xk as V300 = 1.21. Find a symmetric confidence interval for the true mean around the

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