Question

Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization Theorem. (b) Find the distribution of U given T. (c) Let fu\(uſT) denote the pdf of the distribution in (b). Suppose the prior distri- bution of t is t ~ gammala, B) with pdf given by 1 g(1) Tarlé e-T/B, F(a)ßa where both a and B are known. Find the kernel of the posterior density g* (t\u) « fu\-(u|T)g(T). What is the posterior distribution of t given u? (d) Outline a strategy of how to construct a 95% credible interval of 1 given the observed u.

Answer 1

from N(Clo, t) distribution with pafas ryklo) Suppase yir Yar orth be ita rondom scuiple (Y/7) fir ed 20 z YER where peo As Knowon and is unknown

as? (a L91): fort @ The likelihood franchiba Yilt w I e g =T FI van w 21 e 42 - T h (Y) lan e gler) h(y) where li qui po) 4(42=1 Tea and gb7) (31) ly 16 By facetization liescun we get U = x-real is a sufficiens statistic foot

2 llien ? Let x = (X- (es) Yalo = x y=vat flo dy dx = arx Then the pdf of xaso For aya) = 2Jyla C できる上 an Yar!, 1/2 Z X an . x70 120 . x e 2017 Thus, x= (y-(10) n Gamonal or

14 2 Thuis Xi = (Hi-flo? (Yi-Mo)? ~ Gamult 21 ) Then by additive liedzem of gamme austribuhon we qen U = x= -1) Grumman pencer the distribution Un Gavana ( 12 ) ll as

© We know thad un Gammal 1 / 2 then the pat of ll as (1/2) u 1/2 n2t lē - ruin n xolo UB: 770 Suppose the (nor Zistibulion of tas ya Grenna do) with poly laun g(1) = Lotul e B LB are known Then the ternel of posterion densilan and B where both x au 41-(446) as? e greu et Hence posterion disbibulion of 7 8rVenue asn yun Grennial area of

a We lauda Yun Ganuna (340, usp) then 27(81f) ~ J Enrad) 5 7(6+28) ~ vintar) Then 100 (1-0)% credible mulerval fort is given as Please < (UFP)= Mia as) nizal 1720) El- Ne manera M Intro)) = 1-2 (u +25) tetap

credible hulernal Then 100(1-87% for easy Nig, n420) U+33) 2. Nig, na (u +25) then Given, r=0.05 193% credible hulesual for eas No cas, noza), 1975,270) (U128) (U+2B