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Midterm2Solutions - ACTSC 431 831 Midterm 2 FALL 2010 Name...

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Unformatted text preview: ACTSC 431 / 831 Midterm # 2 FALL 2010 Name: ID 5 . The random variable 81 has a compound Poisson distribution With Poisson parameter A1 = 4, and secondaryvdistribution given by f0 = 0.25, f1 = 0.35, and f2 = 0.40. The random variable 82 has a compound Poisson distribution with Poisson parameter A2 = 3, and secondary distribution given by f1 = 0.20, f2 = 0.50 and f3 = 0.30. Also, 5'1 and 5'2 are independent, and S = 81 + 82. (a) Express the distribution of S in compound Poisson form by identifying the Poisson parameter /\ and secondary distribution. .__ {’ _\ g , \L 1 km i: if (> ‘ 1V3, Didi/3M ) £3) \_ ._ r “ , > _ C 2% {INK/i 1/; fl 1 \ ‘1} i f rt“\ } i l / ) P i ' M f a) f i 7 , /-\\ i/i 4/ g :7 V, . ._ -, _ ”T- s 1 _, 1 ‘ ) ‘r ‘ ‘ ,9 ‘ z" ’ i '. ‘ {I i . at: 2 , f _,‘ L _ _.,,,__ N.” x f " " , q f /} ( -p [i . \ - “ i, I i ‘91 X P ‘ (i) i / J i ( 1 / . fl // i f f ‘ f ) \) W ’ ‘} If I ( «. a A I i 5/ fl ,7 0 ‘E 3 ’ 5 (13) Use the compound Poisson representation and the compound Poisson recursive formula to compute Pr(S = 3:) for JD 2 0, 1, 2, and 3. Kw]; 1 W / , f T.\—‘»”\ 1 1/3} ; f 1f, *5 O 1/ / .r ,. i Q L” ® ":76~ LJ’ZJZ/Zq” fgi { \.\ Z g 17' ‘ " w‘ V fl \ 1 ‘ r 4‘3 r1 3 ’ ’ / 1 ‘ " . to ‘ 1 , k ,‘ \5’ '1 Z.) k i‘\ r i 1 1 x x \ o I 1 1 K x 1 1 (W .4 k d * J n 1 l f \ 7 / I ‘ 4) t\ { x) i" I r / I; 1 r 7 w 1, an 1 1 1 A , N (d V \ K \ I K {“(’ L: 1,7 _ A] 41 . J ,. ,,. . : f \ / , few-m Ks. v “2‘ ,’ ’ \r I _ 7 1 1’, m I (A 1 ~ / 1 ~- ” 1 ' c 1 \ 1’ ~ \ “"’ - ' / .e) k f , _, 1 / / “ 2. Suppose that N has the mixed Poisson distribution 009n€~6 pr A n! WW Where 19— (9(6) : —e*T*, 6 > A. F' o (a) Find the lllgf ffo et9b((9)d9. i V .' F ,- ‘1 _ i x, 2‘ 1/ y W; ( j 3/ ' W‘s . F } x, f / ‘y/ w ,1 a ‘{5 X; f f‘k; ( . 0 / (if; C‘ i // J J > k (‘1’ if f/ j ‘ H/ R if ‘ J ‘. 3 (D xx ‘ f"y i:\ ,V 1 i ( r q] /}\\ E . z 1' 3 r g r ( « i A i x J , \, , 1‘ ,., ,J / \, 2 t , ;’ xii/XV) " l ‘ f i K 5 (b) Prove that P(z) 2 SE pnzn : eM5—1){1 — 5(2 — 1)}4. n:O type of distribution has this pgf. 5 (C) Find the mean E(N) and variance Var(N) . w : _j / 3 , ,/ ‘ * ‘ y (“K x‘ ‘ 5' > r"! ,1 /fi L .«' r, f ‘1 :‘ \\ ANN; ' /',“I x’ta“ K \ Explain in words What ft , a \ (d) Show that the pgf may be expressed as 13(3) : €u{Q(Z)—1} ' l ( l / , f n I ‘2 7 ’ A L ¥ \m‘ G) x _ I ll / l \ / r {a Q) /I <3 7 ,‘ 7 x t, i f ’ C‘ ("(7, ~ l i _ ) "r F M \ k i / I 1 f 1 ,2) ( a / / L q) l I l L\ ‘ l r it _ a G W x l (1‘4 K (e) If the number of losses has the pgf in b), identify the distribution of the number of payments, if the probability that a loss results in a payment is v. (Hint: find the pgf.) 7 : r W A: ‘-\, {if l7) 1 1/5/ ‘ 27/ ( l l/ x \_ If C Y e ‘/ 4 ‘y \ /( \ l . \ , V , m , i i ‘ Z r 5 ,\ ,Vfly’flt'filfirl} \‘ is 'J 3. Suppose that Pr(N=n)=(1—p)pn_1; 7121,23,... where 0 < p < 1. Also, let {X1,X2, . . } be an independent and identically distributed sequence of ran— dom variables (also independent of N) with common probability function Pr(X¢=:c)=(l—9)6$; a:=0,1,2,... withO < 6 <1. Let S: 81 +X2+...XN. Use the pgf to show that Pr(S=s)=(1—61)f; s=0,1,2,... where 61 = 9/{9 + (l —p) (1 —- 9)} , I; \‘ \ ,’ W , . \ ,3 \_ or” Jr? , ”R » " “~ - if" H /‘ 1- \X ' 'lf ii i/ : ; “wwm jw-“M' \ E/ / ’_ _, _ f ,_ \\ I, 3;, .. . w _,,' i 7/ and ‘- r ”V“ ’ i , LL . \ , l“ l” I" x ', ., = s ~ . . i 17/ . i I . r / Y / a, ( l \ .4‘ ‘ ’ ' ‘ ( CD M , , 4 Q W / _ m / x \ Q ’ /n/ l V, f r“ J H k R ,",/' -n ,1“ ,1" \ n x 9 r, .. R i ‘ 1‘ “ ‘ ‘ i ‘* 4-, “1’9: )1 s'\w)\c‘mf 57‘ ...
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