7-47. a) f (x) = Then, m+1 m e -(m+1) e - x for x = 0, 1, 2, and f ( ) = m +1 for > 0. x! (m +1) 0 0 ( m +1 ) f (x, ) = m+1 m+x e --(m+1) 0 m+1 0 (m +1)x! . This last density is recognized to be a gamma density as a function of . Therefore, the posterior m + 1 . distribution of is a gamma distribution with parameters m + x + 1 and 1 + 0 b) The mean of the posterior distribution can be obtained from the results for the gamma distribution to be m + x +1 m + x +1 . = 0 m + +1 m+1 0 1+ 0 7-49. a) b) From Example 7-16, = (0.01)(5.03) + 0.01+ 1 ( 25 ) (5.05) = 5.046 1 25 ^ = x = 5.05 The Bayes estimate is very close to the MLE of the mean. Supplemental Exercises n