STAT 514 - HW 12 Solution Key

STAT 514 - HW 12 Solution Key - 7.13.3.9.) _ {(ze)...

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Unformatted text preview: 7.13.3.9.) _ {(ze) :{(i|9)«(9) aficnfi—W/(Zdz)#dé‘wF/fira 1)) Factor the exponent in part a.) as —n(-i,,9')2.__1_(9_.p)2=_ 3. afl‘sx 2_ 1 i_ 2 . 2:2“ 2,2 a“ ‘” -:2:.:27;‘ ‘0' where 6(1) = (r2: +(a'2/n)p)/(1'2 + a'_2/n) and v = (cu/n) / (r + a2/n). Let n(e, b) denote the pdf of a. normal distribution with mean a and variance b. The above factorization shows that _ KM) = n(o.a2/n) wow) = n(6(x).v2} x n(u.r2+02/n). where the marginal distribution of it is 1101, 72+0'2/n) and the posterior distribution of 31:: is n(6(x),v2). This also completes part c). 2 and 8 x: 0'2. Since (n~l)S2/O'2 ~ x34, we have 1 2(el-t)“““””1“‘firm/2923A- ml I((n*1)/2)2(n' V 7.23 Let t = s “Hm z: With «(9) as giveii, We have (ignoring terms that do not depend on 9) ‘ ' -1 2 —1 _ _ . daumkge )I) .e—(n-1)t/29%][Ea;fie-1/fi_9] j“ W ‘mgfcn—nm-a-wMW}, which we recognize as the kernei of an inverted gamma pdf, IG(a,b), with a. = ((n—I)I2j + a and 1 This is a estimator of 7.2" For :1 Observations, Y : ~ Poisson(n)l). a) The marginal pmf of Y is .___....__” °° ( + )—i “137(nii+1) a? p “a - a _ I I . ”.y1r?a)5a ] a A y ° - di ' yimafifi my“) n'T+”'1) _ TM "78 ( ’5‘ 1) Aymara) _ i(¥+a)‘1 in + H a 10!?) --'~n';(-§T“ - M" fl “,0, gamme(y 41:53:19» I‘(y+a) W b) Emy) =(y+a)g%T=;%Ty fmlfi(afi)- 2 mm = (y + calm“?- ‘i‘53 I'm q-s‘ m 2:) Ella Lama—10(9)} m 2m ‘ pg 21 5, c), where z}. n(0,1).$ b) "i'I'l;:'[2C¢Tb""'PuZIs¢;)]=20,b__2(%em¢2/2)t c) If be! > 1 the derivative is always positive since e 5 ' - ' - E[L((p, a), 0)] z E[L((p, a), C) i s <'K] ms < K) + E[L({,u, a), C) Is > K] as > K) = E[L((p, a), C') 13 < K] P(S < K) + E[L((p, 0'), C) 13 > x] P(S > K) = filmy. a).- 031+ may, a). (3)13 > K1P(S > K). 2 °/2<1. when the last equality follows because 0' is 3 if S > K. The conditional expectation in the second term in bounded by Elm». a). ens > x] = E[b 1.4211(0) -. 1C (pus > K] ...-.. spas—15mm > K] >E[2ch—1|s>x] (sinceS>Kand1051) = 2ch - 1 , which is positive if .K > 1/2bc. For those values of K, C’ dominates C. ...
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STAT 514 - HW 12 Solution Key - 7.13.3.9.) _ {(ze)...

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