Question

Random variable X corresponds to the daily number of accidents in a small town during the first week of January. From the previous experience (prior infor- mation), local police Chief Smith tends to believe that the mean daily number of accidents is 2 and the variance is also 2. We also observe for the current year the sample number of accidents for 5 days in a row: 5,2,1,3,3. Let us assume that X has Poisson distribution with parameter θ . Using the Gamma prior (hint: suggest values for the parameters of prior distribution using Chief Smiths previous experience and formulas for the moments of Gamma distribution), determine:

(a) the posterior distribution of the parameter θ given the observed sample;
(b) according to Chief Smith, the Bayesian estimate of the parameter θ (posterior mean);
(c) ignoring the prior information, the maximum likelihood estimate of the parameter θ.

Answer 1