The number of defects in an iron casting can be assumed to follow a Poisson distribution with mean 0. A quality engineer inspected nine castings and observed the following number of defects in them: 0, 2, 2, 3, 3, 1, 2, 1, 1. Assume that o has a prior distribution Gamma(2, b), where the hyperparameter b is assumed to have a distribution Exp(1). Xi, Xiwin Xn ~ Exp(A) Et-1 x; ~ Ga(n, A) f(x|2) o an * e-dix 1 ~ Ga(a, B) n(A) o 20-le -B.A Posterior: n( 1|Ex;) ~ Ga(n + a, Ex; + 8) My answer so far . Prior 8 ~ Ga(2, b) Ex = 0+2+2+3+3+1+2+1+1 =15, n = 9 Posterior: n (1|Ex;) ~ Ga(2 + 9, 15 + b) = Ga(11, 15 + b) Posterior Mean: E(1|Ex;) = 11 15+b Posterior Mode: Mode = 11-1 = 15+b Posterior Median: (Numerical: gaminv (0.5, 11, 1 = 15+b