[30] A muddy pond contains an unknown number of 0 fish which are uniquely tagged with integer numbers 1, 2, ...., 0. The experiment/survey to collect data for estimating 0 is to be conducted as follows by: n fish will be randomly caught one at time and returned to the pond after recording their tag number represented by the random variable X. This is random sampling with replacement. The resultant random sample of the tag numbers of the caught fish is represented by X1, X2, .... Xn, and each X, has a probability mass function: if x = 1, 2, ....0, f (x10 ) = 0 otherwise. Given: E(X) = - 0+1 - and Var (X) = 02 - 1 2 12 (a) Prove or disprove that 0 = 2X - 1 is a method of moments estimator of 0. (b) What is the mean and the variance of 0 in part (a)? (c) Show that 0 in part (a) is a consistent estimator of 0. NN WEN (d) What is the bias of X = - > X; in estimating 0? n 1=1 (e) Prove that X is an inconsistent estimator of 0