Answer Happy

3. Suppose that X1, ..., Xn are independent samples from a Poisson(1) distri- bution. For reference, the Poisson() probability mass function is e-tk P(X; = k) = (k = 0,1,2, ...) k! and the mean of the Poisson() distribution is .. == (a) If we observe X1 = x1, X2 = 22, ..., Xn = In, what is the method of moments estimate for X? What is the maximum likelihood estimate for X? = = 12,..., = 210-le=8'3 (b) A Bayesian statistician proposes that a good prior distribution for X is Gamma(a, b) for particular fixed values of a,B. Thus, the prior pdf is Ba f(2)= x ja-e-B1 (4>0). r(a) If we observe X1 = 21, X2 Xn In, show that the posterior distri- bution for 1 is Gamma(a', ") where a' = a +21=1 li and B' = B +n. Hint: If you can show that the posterior pdf for is -e-8'N (4>0) for some normalizing constant Z, then you are done, because the posterior pdf must integrate to 1 by definition and this forces Z to be equal to the correct normalizing constant for the Gammala', b) distribution. So you can get away with ignoring all constant multiplicative factors, where "constant” means any- thing that does not depend on ), as long as the posterior pdf ends up being proportional to je?-12-89. To start with the prior pdf, there is no harm in saying f(1) 19-1e-B1 since the "missing" multiplicative constant Bº/T(a) does not depend on ), so in the end it would just be absorbed into the normalization factor Z anyway. (c) For a Bayesian point estimate of 1, let's take the mean of the posterior distribution. How does this compare with part (a) when n is large?