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ACTSC-432-1055-Quiz_exam

# ACTSC-432-1055-Quiz_exam - QUIZ — ACTSC 432/832 SPRING...

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Unformatted text preview: QUIZ — ACTSC 432/832, SPRING 2005 Family Name: g Given N ame:‘#g_ ID NO. - Date: Wednesday, June 1, 2005 Time: ' 4:30 — 5:20 Total Questions: 2 Total Marks: 50 Total Pages: 9 (including this cover page and a 3~page formula sheet) Aids: Non-Programmable Calculators INSTRUCTIONS 1. Print your name and ID. number in the space provided on this page. 2. Full solutions are required. Part marks will be awarded for partial solutions. 3. All your work should be done in this exam book. You may use backs of these sheets as scratch paper. 2? r8 7; 17L; / 5‘17 QEL_________________________._____I£g_C;2 k6 Si 0 1. Let Xj be the number of clai sin year j. Given 6 = (9(0 < 6 < 1), X1, ...,X5,X6 are independent and have theéme'binomial distribution 61419). The distribution of G is a beta distribution Beta(1, 3). (a) Calculate the variance of the number of claims in year 6. [10 marks] @ r v w'n \“V VW (XL) : EWM lXth] ‘l VML l" (Xelxrw‘ = EMeu—exjwmme) Emmi—Z; :‘ 4 E ( (a ~91) +~ mm; VWl= Wig -—~ am; A f _, ' .. {I l 7‘1 ("Eff/E I all W , , ‘ r: 2. 4 [my gicx’j “t [tn OMS mm = a m - a low; If; ‘ q “‘4 =11 / . €[Q1‘JVZOJ /0 (b) Calculate the probability that there will be no claims in year 6. FVQXQiO) :(,. PYQXE M’P Q ,1) R” ”t J!“ ' 4,, WW 1 0 <§>BWW~ mm WQQ Q) (Q9 Q30 _ L}, FQX’H3PH’X) , 3 W W) : 1H ﬂw)‘ : 2a {wax EQWX)‘. ’1‘ 50% E‘H)‘. X (77 ‘ I z ‘ w Luv" (3/ _, N 770 l / WW q 513% ’4‘" ’mo 2%: 1’72” 7 7< W X m f“ (9» 27 g f (’2’ page 4 1 Quiz page 5 .Let X- be the amount of Claims in year j Given @- — 6 > 0 X1. X2. .X10, X11 are 2independent and haxe the @conditional probability density furictioh as f (xf6)_ J164/1" if :r > 6 and 0 otherwise Furthermore. suppose that G has a uniform distrib- V} \f ution U (5, 50). C1a1ms in years 1- 10 have been observed and their sizes \$1, ...,\$10 were KW . 15, 23, 10, @119, 60, 90, 45, 65, 55, respectlvely. (a) Determine the posterior distribution 7r(6|:1:1, ..., x10). [10 marks] 6 :41 , 4 1 K(®\j\|\“‘j\10>®)\ 3 . Sgex 3:) O I PIS/51111817111171 , 49! 13E.\ 1 “9 11‘ 16 S “q: 6L” 7 , 745 @161 I ”11’” s ﬂ 717” 7‘7 ’ f 1 111me _, 551172711 ,1. “C éthSrbgmiu 11111111 - 1 vao 0/ 7111111, 11111 we“): 1119/10 ”6 ...
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