Question

have a Ba- Question 2. (20 points) Let (X1,...,xn) be a sample from Poisson(2), and let prior distribution Gamma(a,b), a,ß > 0, with pdf fe exp(-82), when 1 2 0, (A) =1 Ta) * 10, when 1 <0, where I(a):= 6°40-1 exp(-t)dt, for a > 0. (a). (10 points) Find the posterior distribution of 1. (0.1) (b). (5 points) Calculate the posterior mean and variance. (c). (5 points) Now consider a Bayesian test of H: 15 lo versus H:2 > 10. Calculate expressions for the posterior probabilities of Ho and H.

Answer 1

ܡܦܥܲܠܟ aquen that *1982 a wong wild poica) and an Granma (@aB) with pdf T(2) =( Ba 20 eXP(-B2) ,aso a >o e fortere blessity fuerieties of the ris. *19*2"Xo quien that is given by . T** (212) a f (x 1px 29 cm Xn) x^(2) a ↑" [exp(-x) - 2² kil] Ba 20- 1=1 - exp coBoc kitara o exp(-32) a exp(-a ). 2 a exp 6-7 (A+B)). 2{xita-a - [laz ~ Gamma (E2i +2 g h + B)

I b) Posteria mean E (218=%) = { ifitas sf xoo Gammald, B7 + 8 L E (8 ; 0 ; C = 7 ܛܒ݁ܶܥܵܬܐܠܬܘ ܬܠ ܐܥܥܠܐ was calx 20) = { "xito lona B ²