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Problem 5. We have access to a file consisting of n = 10 numbers. The numbers are either 1, or 2, or 3. Moreover, the value 1 appears n1 = 2600 times in the file, the value 2 appears n2 = 5200 times, and the value 3 appears n3 = 2200 times. We know that these numbers are generated i.i.d. according to an unknown distribution. (a) Let / denote the mean of the distribution. Estimate the value of / from sample data provided in the file and provide a 95% confidence interval. (b) Assume now that the generating distribution of the data has the following form: with probability P1, X = 2, with probability p2, 3. with probability 1 - (P1 + P2) We would like to estimate the value of the parameters p, and p2. Consider the following estimator for the value of PI(X1, . . ., Xn) = -ER{Xi = 1}, n where 1{A} takes value 1 if A is true, and 0 otherwise. Compute the estimate pi from the sample data provided in the file. Is this estimator and unbiased estimator for pi? Justify your answer.(c) Use the method of moments to estimate the value of 331 and 192. (you should compute the estimates from the sample data) (d) Now, assume that the precise value of p1 is given as p1 = :11. As a result, we now know that the distribution of the data has the form: 1: with probability 41, X = 2a with probability P2: 3 a with probability 3 P2 We would like to estimate the value of the parameter p2 from data. Find the maximum likelihood estimator for p2 and provide a 95% condence interval