Question: without any software help (R/Matlab or other) Consider a random sample X1, . . . , X1n from a distribution with the probability mass function

without any software help (R/Matlab or other)

Consider a random sample X1, . . . , X1n from a distribution with the probability mass function 62 2P X2 26 3P X3 1 (9+1)23 f()_ I'( _ )_(9+1)2: f()_ I'( _ )(6+1)2' where 6" > 0 is the unknown parameter. in) = Pr(X = 1) = (a) Calculate the mean of this distribution and use it to derive the method of moments estimator of 6. (b) Let Nk be the number of variables X1: such that Xi = k, for k = 1, 2, 3, with N1+N2 +N3 = n. For these data, calculate the maximum likelihood estimator of 19, :9. Show that the likelihood is maximized at 6 = 9. (c) Find the CramerRao lower bound for the variance of an unbiased estimator of 9

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