1. Let XXX, be a random sample from Poisson (0), where is the unknown parameter. Assume the prior -8 e density of is given by (0)=- 0>0 0 otherwise .Derive the posterior density function of given that T=t where T=X, and hence find a Bayes estimator of i=1 -Ox 2. Suppose that X, X, X, are ID with common density function f(x:0) = {000 17-vis where (>0) is the 3. otherwise unknown parameter. Assume the prior density (0) = {ae [aea 0>0 ' , where a (>0) is known. Derive the otherwise posterior density function of given that Tt where T = X, and hence find a Bayes estimator of i=1 An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with unknown mean and a standard deviation of 100 hours. Assume a normal prior distribution for with a mean of 800 hours and a standard deviation of 10 hours. If a random sample of 25 bulbs has an average life of 780 hours, find the probability that the average life of bulbs is between 776 and 813 hours.