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347.2007.final.sol

# 347.2007.final.sol - NAME 547\$ U T a/Vj ACSC/MATH 347/447...

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Unformatted text preview: NAME 547\$ U T/ a/Vj ACSC/MATH 347/447 Final Exam December 17, 2007 Directions. . Show all your work. Full credit will not be givenfor anyproblem unless sufﬁcient work is shown or a suitable explanation is given. . You may not use the cdf or pdf functions on a calculator in any problem. . Part of the credit for each problem will be for style, readability, and mathematical correctness. . The maximum score is 100 points. . 21/2 hours is allowed for the test. 1. (5 points) An urn contains twelve red and eight white balls. You draw six balls from the urn without replacement. Let the random variable Y denote the number of red balls drawn. Find the probability that Y Z 5. Y ﬁdel— a/ /W MM’WZ’ZKAL M1 ale) «at; We , (L: e .. m; We ft—ﬁfMW' rec/2;) ,3 PM?!) + szw/ .. 72 if) 72' 8 (779%?) WNW!) " (in 7* (if) = W7 ‘sww é (a; '23; 0a [273 2. (10 points) You play backgammon regularly with the same opponent. You are a better player and win 75% of the time and lose 25% of the time. (You can assume that the games are independent.) a. If you play 10 games tomorrow, what is the probability that you lose exactly 2 games? [Wt Y = ﬁﬂcmvi’e-W 6e“ ﬂew/Mum M. \(rcéeaﬂejoamrjj 45/” PM = M: (QMawls’fwﬂflg: wig/e b. What is the probability that you play atleast 7 games before your ﬁrst loss? ACSC/MATH 347/447 Final Exam December 17, 2007 Page 2 of9 2 . . 3 , 3. (12 points) Let Ybe a continuous random variable with densnty function f(y) = y 0, elsewhere. OSySl a. Find P(Y >3 b. Find the mean and the variance of Y. (GUN (5% it}; :oerwcﬁ‘; fZMy :04 ‘4 a 0Q 5L _ «L ,_ ,1 gorlﬁaov W4— lg (WW/\$2 7, 264:1- 2535—1-07“ % 5 a m- vm= 5(7)“?le . l 49—Lﬁ5’;,.3, '5 %’lﬁ3?) = 86 2 ACSC/MATH 347/447 Final Exam December 17, 2007 Page 3 of 9 4, (10 points) The failure ofa circuit board interrupts work that utilizes a computing system until a new board is delivered. The delivery time, Y, is uniformly distributed on the interval 1 to 5 days. The cost incurred following a board failure is related to the delivery time for a new board by the formula C = 50 + 3Y + 7 W. a. Find the probability that the delivery time, Y, exceeds 2.5 days. b. Find E(C), the expected cost associated with the failure ofa circuit board. (Cl) Y NM[1)5))M ‘ ,__ 5'3-12105" gaff" 1"” l’(\/>Ra§) — 5___ :. 4 a: aﬂoat; 4-. we) Em: E‘ts‘?+37+ We") '1 97) + 35(7) 1* c; 9(sz , \$ ACSC/MATH 347/447 Final Exam December 17, 2007 Page 4 of 9 5. (10 points) A discrete random variable, Y, takes values in the set {0, l, 4, 9, 16} and has probability function given by the table below. a. Find k. b. Find 5(77) . 4% 0 | 1 I 4 | 9 16 | p0) 0.10 | 0.35 | 0.20 | 0.25 k | one) + 04352,; on }o-+w QF-f}; ”=71 i2, : 0M0 /——"—"—M (4) Em?) 2: EWWV') 7.. ®(0\/‘0)-+ Jiwas’) +\/L7{0‘w10) + \l? (0015’) + W; [045”) : o + a. 35' +0.. 75+ 0, 7; 1" ave/’0 .5 10.95 ACSC/MATH 347/447 Final Exam December 17, 2007 Page 5 of 9 6. (10 points) Bean seeds from supplierA have an 85% germination rate and those from supplier B have a 75% germination rate. A seed packaging company purchases 40% ofits bean seeds from supplier A and 60% from supplier B and mixes these seeds together. a. Let G be the event that a seed selected at random from the mixed seeds will germinate. Find P(G). b, Given that a seed germinates, ﬁnd the conditional probability that the seed was purchased from supplier A. (CL) 3% 19044 20(403 PCB): Chan P (Gt/Jr) :0: 8st,, PUP/5910a 7;— ACSC/MATH 347/447 Final Exam December 17, 2007 Page 6 of 9 7. (10 points) The random variable Y has a normal distribution with mean 3 and variance 4. In each part, show your work or explain how you get your answer. Use the normal tables, not a calculator. a. Find P(2 < Y< 5). b. Find the value c such that P(YS c) = 0.9960. Y’V/Vhr‘r) W"E1 7::‘2 N/Vfal,i) ) (k c Q~ , XW 5674 We, [>(;\<\{<'.s*) '5 P(“6’~5‘<ZL:L) cu 249-7124 we) 2%; <1)- NZ >005”) :. p(%4.i)_ii _P(%55‘5‘3) 0’3 ._ /‘ / ’79:“ " 024% 2 W C :1 3+}(O'Z‘625‘) ~7— y, ACSC/MATH 347/447 Final Exam December 17, 2007 Page 7 A— 8. (15 points) Let Y1 and Y2 be continuous random variables with joint density function 2y1+y2 f(y1,y2)= 4 , 0<y1<l, 0<y2 <2 0, elsewhere. a. Find P(Y1 + Y; s 1). [Draw the picture!] b. Find the marginal probability density function, f1 02]), of Y1. c. Find the conditional probability density function, f (yz I i), of Y2, given Y1 = i. (or) WWW; e 41> ;_ {3' ma) M l: ACSC/MATH 347/447 Final Exam December 17, 2007 Page 7 B 8. (15 points) Let Y1 and Y2 be continuous random variables withjoint density function 2y] + )2 f(y1,y2) = 4 ,0< <1, 0< <2 . yl .V2 2 0, elsewhere. ‘ _ a. Find P(Yl + Y2 s 1). [Draw the picturel] ’ 4 b. Find the marginal probability density function, fl (yl), of Y1. 0 a1 ‘ 1 c. Find the conditional probability density function, f (yz I Z ), of Y3, given Y1 = i. (7/) (mﬁﬂw/ ) ACSC/MATH 347/447 Final Exam December 17, 2007 Page 8 of 9 9. (10 points) Let Y be a random variable with density function f (y) = { 2% dis:w}}11;e1 Find the density function of U = 2 — 2Y. I I am“ Z/%WM5/W Oéﬁz‘il 0>‘12y7'2~ ai>u=2~~2#>o gtﬁ::aii::f‘e* déqeec m'W/g/oﬂém W= yam-Mu) Fu(u):P(UéC{): P(&‘9\‘/£M) :P{~27£ M'L} =WCY>1~%):fLaydﬁ 4’1 174% = U .. L- ' ”L- r 0 _ ii 4 ) F," [(1) 1 ) (X \C C U 1—(‘1—437‘ 0<M“Z I .L/ (A 2% M 799(17): a-l(1~%)(“'i)=1"‘§~> 0"”‘9‘ ACSC/MATH 347/447 Final Exam December 17, 2007 Page 9 of 9 10. (8 points) Print requests arrive randomly and independently at a laser printer on a local area computer network. Let Ydenotc the time in seconds that it takes to process a request, and assume that Y is exponentially distributed with mean l0 seconds. a. What is the probability that the next request will take more than 30 seconds to process? ”W70 vamp/ob M— Fly) a. 6 7%) ~ ewf7H0 Slaw—8 )vﬂ>o ‘ g - // P(‘/>30)= 1- brawasraa): 6 30 0,3“ 2.090??? \ b. Processing started on the request currently being printed 15 seconds ago. What is the probability that it will be ﬁnished xx ithin the next 10 seconds? bygﬂmewa4 P(Y4;S~I \/ LLB/5) 5M ﬂ—Q WWW Mir—54M r’L/ﬂwﬂj/‘ég- ﬂux—e...’ Wage/rug; WW ' _m//0 _;, Nye/a): I=(/c2)=:L—e 21—52 2 00 6392) 11. (8 points Extra credit) Suppose that Y] and Y; are random variables with E(Y1) = 2, E(Y2) = —7, V(Y1) = 2, V(Y2) = 4, and C0v(Y., Y2) = A ‘- Find the mean and the variance of U= 6Y1 — 5Y3. gm: glam 5w) aﬂvJ—s‘glw 3%) )[;.)=~(\$‘)(—7): 47 Vm): V [é Ypf‘lﬂ (31w x.) H'Ffvmlmw)(-5)Cwl‘/u‘/2/l 43(9qu HMXLF) e 6061) 17a T/O’O~ 1510 2% ...
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