Question: Let X1, . . . , X9 be i.i.d. random variables following Poisson(). The PMF of Poisson() is P (X = x) = xe x!

Let X1, . . . , X9 be i.i.d. random variables following Poisson(). The PMF of Poisson() is P (X = x) = xe x! . The mean and variance of this distribution is E(X) = , Var(X) = . (1) (5 points) Find the likelihood function of . (2) (5 points) Find the maximum likelihood estimator (MLE) of (denote it by MLE). (You don't need to check the 2nd derivative.) (3) (5 points) Calculate E(MLE) and Var(MLE). (4) (5 points) What does MLE converge to in probability, as the sample size n go to infinity? (Assuming that X10, X11, . . . , Xn also follow i.i.d. Poisson(). State any theorems you used in solving this

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