Question

iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show that the posterior distribution of 0 is Gamma(nTk, n ). (b) [4 marks Find the probability function of the marginal distribution of Y = nX. (Note that the conditional distribution of on Y is not the same X1, ..., Xn.) as on

Answer 1

Sol ASSume that O Gamma (k,x) and X, .. .. le-eid dnota 1 Resuired infoimation is given by Xi Here - Knle=0 Apoison (e), -therefoe the SinCa Xi,X2 ,X3. Likelihood unction is given by LK, X2,X3le ] , nle)] L [x,x2, 3 Xi >d) a) shw that the posterior distribution of e we have to obtained the pasterior listiburtion of Gamma [nrt k h+a] is, Here palf is given by The ek-leae Ple) k b) bave to taken eyn@ and coeget the posterior slistrikution of e is given Observations, Can be obtained as the, we X, -Xo fle) Ixi

dOstn)-ot)e and the gamma distribution -im is osed to rramekr (ktnx) and (ntA) that means O(x,,x,X --AnJu Gamma ktnx, t there fore. we have Stowed that the posterior oli8tribution of 0 (b)firnd the ptabilty furction of the margina! clistribution haue to obtained the podoabikty fnction af mavginal clstibution is. Here SinG X,,X2,XK |O~ poisson (e)diStributon is mayginal enaating-function is, of x is, Ce et- that we manginal et) E etx] mean mavginal distvibution is [xtx2tX3+ +X,] y Therefoie C) the maginal distribution geneating fnchio is, of y an be obtained a

Tety] et atetott] E letn, et,ty E etElet Elet [etx)J oala E Ee FO LE E let-11n enefet- herefore poolaability funchon of the marsginal dishi bution axhudive pron fyt)ene(ne) e aiod pota csha bonlo