Suppose that X, ..., Xn are independent and identically distributed Poisson(X) random variables with probability function...

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Suppose that X, ..., Xn are independent and identically distributed Poisson(X) random variables with probability function Define X and S as f(x; ) = = n (a) Find E(X) and Var(X). (b) Show that Xe-1 x! X and x = 0, 1, 2, .... XX. = X. (c) Show that E(S) = X(X + 1). (d) Let D = S-X. Show that D is an unbiased estimator of X. (e) If = 1, use the Central Limit Theorem to find an approximation for P(X + X + + X100 90). (You may express your answer in terms of the standard normal cumulative dis- tribution function .) Suppose that X, ..., Xn are independent and identically distributed Poisson(X) random variables with probability function Define X and S as f(x; ) = = n (a) Find E(X) and Var(X). (b) Show that Xe-1 x! X and x = 0, 1, 2, .... XX. = X. (c) Show that E(S) = X(X + 1). (d) Let D = S-X. Show that D is an unbiased estimator of X. (e) If = 1, use the Central Limit Theorem to find an approximation for P(X + X + + X100 90). (You may express your answer in terms of the standard normal cumulative dis- tribution function .)

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