Given, .... an follows poission and the prior is Gamma (ox, $). then the posterior also follows Gamma distribution with parameters. ' = sum of x : and B's B3 hB+1 ( x , B ) = ( 2, 1/ 4 ) Thus, in this Case d' = 2+ (3+1+4+5+2+3)- 20 B's 1/4 = 1/10 6x 1/ 4+1 Thus the posterior is Gamma (20, 1/ 10)Given, .... an follows poission and the prior is Gamma (ox, $). then the posterior also follows Gamma distribution with parameters. ' = sum of x : and B's B3 hB+1 ( x , B ) = ( 2, 1/ 4 ) Thus, in this Case d' = 2+ (3+1+4+5+2+3)- 20 B's 1/4 = 1/10 6x 1/ 4+1 Thus the posterior is Gamma (20, 1/ 10)a) In a sequence of consecutive years 1, 2, . . ., T an annual number of bankruptcies are recorded by the Central Bank. The random counts Ni, i = 1, 2, ..., T of bankruptcies in a given year are modeled using a Poisson()) distribution and can be assumed independent from year to year. The Central Bank has collected data for the past six years and recorded the following counts: 3, 1, 4, 5, 2, 3. The prior on A is a Gamma(2, -). Determine the posterior distribution h () |NV). Note: You may use that for a known a > 0 and B > 0, the Gamma(a, B) density is given by: f (x; a, B) =- r(a) Ba ' r >0, where r(a) = is the gamma function. Also, if X is distributed Gamma(a, B) then E(X) = ab and Var (X ) = ap2. O Gamma ( 10, 20O Gamma ( 10, 20 O Gamma 10 * , 20 O Gamma ( 20, 10 O Gamma ( 20, 10h] The central bank {from part [a]} claims that the intensity A is less than 2. The central bank would like to test the claim via Bayesian testing with a zero-one loss. Determine the value of the posterior probability required to perform this hypothesis test. Give your answer to 2 decimal places}. You might want to use the following: I'[n] = [n 1)! and 2 f Ewemmda: = 0.000644 [:I