5. (39 points) Recall that a discrete random variable Y is said to be Poisson with parameter 9 > 0 if the density {also called probability mass function) of Y is Recall further that if Y is Poisson(9), then lE(Y) = 9 and Var(Y) = 9. (a) It turns out that the same formula for the Fisher information can be used for discrete random variables. Show that if Y is a Poisson random variable with parameter 9, then 1 I 9 = -. ( ) 9 For the remaining parts of this problem, suppose that Y1, Y2, . . . , YlriL is a random sample from a Poisson{9} population. (b) Show that 9M0M, the method of moments estimator of 9, is Y1++~+Yn n . MOM = ?= (c) Show that 9M0M is an unbiased estimator of 9. (d) Calculate Var(9MOM). (e) Explain why 9MOM must be the minimum variance unbiased estimator (MVUE) of 9