I'm stuck on this problem and I was wondering if someone could help me out. My initial guess is that if you set e^-cY equals to e^-theta => you'll get that e^-c(ntheta) = e^-theta => c = 1/n. However, I'm not sure that this is right for a. I have no idea how to begin B and C, but I know that 1/n goes to zero as n -> infinity.
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Suppose that X1, X2 , ... , Xn form a random sample from a Poisson distribution with unknown mean θ, and let Y = ∑i=1 Xi.
A) Determine the value of a constant c such that the estimator e-cY is an unbiased estimator of e-θ
B) What is the lower bound for the variance of the unbiased estimator found in part (a)?
C) Suppose that we wish to ...
The solution involves the unbiased estimator of Exp[-theta] of Poisson distribution.