Spring 2010 Exam 2 Solutions

Spring 2010 Exam 2 Solutions - Probability and Statistics...

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Unformatted text preview: Probability and Statistics for E-ngioeet‘tog (ESE 326) Exam 2 March 18, 2010 This exam contains 11 multipieuohoice probiems worth two points each, seven tmevfaise probiems worth one point each, six shortenswer probiems worth one ooint each, and one {tee-response probiem worth five poinis, for an exam totai of 40 points. Part L Muitiple—Choioe (two points each) Cieetiy fifi in the ovai on your answer card which corresponds to the only correct response. E. Consider the continuous ranéom variebie X whose density is given beiow. Find the median of X. f(x)x§x~1?2=4—17; 1gxgg ((A) e E (8) 4 (C) 4 )1 ’ (ms éxlxi 9C :1 £54 (E) 5% (F) 5% (G) 6 (H) fi (I) 2x5 (J) 4\/§ 2‘ Larry drives a truck each. week from a distn'bution center in St. Louis to an outlet store in Lake of the Ozarks. The amount of gas he uses on this trip (one«way) is normaliy distributed with a mean of pt 3 18 geiions and a standard deviation of .8 gaiions. What is the probability that Larry will use less than 17 gailons of gas on this week‘s (one-way) trip? (A) .o751 @ 4.1; (C) .2113 H jay}... "1/1, léyzix Log/Emit « 3— Q‘JZ- 013} I 2‘ 2x “2. t. 2- E 3 WIN WU“ (D) .2283 ; g: E? (E) .3364 E?" (F) £636 gamma; (1.. was) m) 3%} 5% e gem (o) .m? (H) 3388’? (I) .8944 (J) .QZéQ Larry's bess would like to budget the trucking company’s expenses more carefully. fie asks Larry te determine the gallon amount such that 95% of the Kips (described in problem 2) use this amount of gas or less. Please help Larry work this out. (A) 16.1 i (B) 36.4 § , 1 . (C) :63? i? .g (1)) 1m F V (E) 19.0 fig { 19.3) w emfémm Klee) 1793,?) :3 5e; 3 (G) 19.6 (H) 19.9 At a certain casino (which is epece 24 hours a day, seven days a week), a: yartieular slot machine delivers a large payout on the average every 14 hours, at which poim it draws attention to the fact with loud bells and home. Assume that this is a Poisson process. Denise starts playing this slot machine at midnight and continues pieying all day. "What is the probability that she gets her first large payout between 8:60 am. and noon? E M (A) .1014 em 1 : 1% (B) .1089 .fi/M Cm) "1493? We} "4' i» e (D) .1822” Pf? f X 5: a]; a; F’ée‘) » F762?) 25>??? (E) .8178 w w $.31 A? y “ gig/jg @3597 “(We fi>“é/éee; ) (G) '89:} ': avg/fie a: ah 52"} s {4633 (H) .8986 ' The time to failure 0f a car battery is a Weibuli random variable with a m .03 and {3’ = 1.5, Where time is measured in years. find the probability that. the battery will fail sometime during the first SGVEB years. a ti‘g’ (A) .6683 Riel r: E ‘ «z; 7e (B) .1191 a _ . k _ 7 m£03(79.§) (0.1666 @[Ye‘fliif'flzzmfifflzhg 1‘59 a 422;: 3 (F) .8334 (G) .8809 (H) .9317 6. The foliowing diagram ShGWS the reiiabiiity of each of the fear (independent) camponengs in a system. Find the {eiiabflity 0f the Whole system. (A) .48? (B) .513 (C) 156825 03) 5’41 (E) .922 (F) 950% (G) .9652 M (H) 974925 (I) .987525 (I) .99975 I i -‘ 6136‘535’) :— fl"? ~— (ay (1233 if ,‘f‘iS' mflz’fevfl ,%§2§’ Consider the two-dimensional} continuous random variable (X, Y) whose joint density is given as foIEows. Set up the doubie integral} neefied to find PLX > 1 and Y > 3]. E Jim/(32', 31) = $1“? (A) f: f: é; my dy dx (B) j: I: 3%ng 0531 six (C) I: 5 gig $3; d3; d2: (1)”) fff: “$373; dy d3 (E) f: 1: 33% my 51;; d2: (F) f: : gig my dx 0131 (G) _Kfiz§§xyd$dy (H) fjffgg-zy dz dy .. 4 4 (I) Eglixydxdy {3) fidfyzlgxyda‘: d3; 0<x<y<4 Consider the foiiowing theorem am} proef. in winch step of the proof is the assumption of independence used? (Note that a “step” is a transition from one line to the next. For example, choice (A) wouid mean that independence is used to justifi’ the transition from fine 1 to fine 2 of the proof.) Theorem. Let (Xg Y) be a Woudimensienal diecrete random variable. if X and Y are independent, then r m: EEXYI u (A) _ g e A = ZZfofleW "W W m Wieweeme y {e (e); éj X’ W“? 94¢ 4;V{E,j3: fiixlfifi» mggxymxmo) gm x Wig; a (c) m 32;: {we}; meme] 8 (a) = [gxme] [gymw] z; (22) mE[X]E[Y} Consider the two-dimensional discrete random variable whose joint density is given by the following tabIe. Find Cov(X, Y). (A) ’41 E’fx] 2: (33(33 4* {4‘30} 2: 337 (B) —.09 f _ I m ((3) _K05 ELL?) 3 ‘3“ 52352155” idle E’fxefl 22’ (3%») e {o )(1 2A 9» [#35 3% (a? 3!; if) (E) a _ (F) .02 ‘3 5’3 (G) “35 cwzxy) :2 Sf?” (3:?)[som «*3 Mia}: .99 (I) .11 1;) 10. Consider the two-{iimensiona} cantinuous. tandem variabie (X, Y) with joint density and magma} 1}. densities as sham bebw. Set up the integrai expression which would be needed to compute the curve of regressian {Lygx (1’? Y {m X. fxflxgwm §§{x2+y2} 05x52 0$y§2 fflxjm %5(3xg+4) {)‘<x£2 fy(y)mg%(3y2+4) 0-2132 gammfi 2 in f X Z “3.,- (A) I;de g {g > a ’M‘vaxjejE w 33 {:21 w?) , {"3" £3} “ ”” ‘ ; imfimj} 5?: k”; g; {x}; g‘Tfi/Exinw 2 q (3) fl) "3‘1—1(3yz+4} ‘5‘“? 2 (B) fa Z 2 §-${_3':2+' 2) : g (F) f9 Wig - a % WW 2 ifixfiflz) (H) In W4? In the movie Blacking and Samgiing, the technique cf blocking is first described as the host, Teresa Ambiie, sorts which of the following? (A) baseball cards (B) CD9. (C) eggs (D) gemstones {Effiundzy (F) LEGO-blacks (G) magazines (H) marbles (I) (D recyclabies shoes Part II. True-False (om: point each) Mark “A” on year answer card ifthe statement is true; mark if it is false. 12. Fer the continuous random variable X whose deesity is given (and pietureé) as fefiews, P{X : 2} z .75. f(x) = ~.75(x — 1)(:z: w 3) 1g x g 3 ‘3 2E “'3 fl 3 aims-‘1‘- The graphs beiow are densities for continuous ranéem variables, f1 for X1 anti f2 fer X2. Use these fer problems 13%4. 13. [£351 > {3X3 1W PL” «its web, 14. Viier > Vang )5. he. Wee XM. 15. If X is a continuous random variable with [ax m 25 and 0X 3 1, then H23 g X g 2?] m .95. i" 5% W4) 21;) X is rmmwfm $352,3£Xé2”?333f%§ 16. If (X , Y) is a two-dimensionai continuous random variable with joint density ny(x, y), then 00 f;:f:$fXY($yy)dyd$ : meffizv) dz. imam m ireéie MWMQW 4 £7. Let (X , Y) be a twe-dimensienal (discrete or eentinueus) random variable. If pr m 0, then X arid Y ’é- d t. ‘ v " ‘" erem epen era wbfl/ if X W ‘2‘; ms. kWh/l} W 155;“? 1G! 18. Let (X, Y) be a Wofiimensiona} (discrete or continuous) random variable with joint density fxfim, y). IfX and Y are independent, then fylfl; (y) 2: fy (33). em feisfi 3., JWW *3 a. f {.3 3 5’ am i We ‘ w “i” m: 3% 3mm aj fie gWiéfi—éfifiilflm i’art Iii. Short Answer (one point each) The answer to each of these is righi: or wrong: no work is required, and no pertiai credit win be given. Give only one answer to each. (Ifyou give more than one answer, the poorer one Wiil count.) 19. Let Fix) be the cumulative distribution fimction for a continuous random variable X. Then lim F(a:) = i memo 00 r ’ i n 20. What is the exact vaiue of the following integral? f0 z“1fge”zdz 1:: PEB ~ \E'Tr 21. The iength of time to failure of a certain system is a Weibul] random variable X with the foiiowing fsiiore density, Is the hazard rate increasing, decreasing, or remaining constant over time? Hm) m éxwir’ige'zm 3: > 0 2‘ Answer #22 and #23 with one of the words, “always,” “sometimes,” or “never.” (Note that an always- sometimes«never problem is somewhat iike a true-faiso problem, where “eiways” corresponds to “true,” and ”sometimes” and “never” are more~specifio versions of “false.” ) 22. If (X, Y) is awe—dimensional (discrete or continuous) random variable, then Var(X + Y) m VarX + VarY. 23. if (X, Y) is a twondimens-ional (discrete or continuous) random variable, then —1 3 pm» g 1. 5‘ 12,11;le 4..» 24. Consider the halo—dimensional discrete random variable (X, Y) whose density is given by ihe foliowing rabie Fifi in the table for the conditional density fygxmg for the random variabie le : 2. y = 5’ g re frown): j...» 3,. g... 5 5’ s: Part IV. Free Resggnse (5 points) Foliow directions carefixfly, and Show 33 the steps neeéed to arrive at your solution. 25. Consider the {woudimensionai continuous random variabie (X , Y) whcse joint density is shown beiow. ’7 fiw($,y)=§% O<x<3 9<y<oo Find P[X S 2 and Y Z 10}. “ 'g‘L-i‘ w%(w-w‘ 4’95; ”* bk})@ 3 £3 f, I: :1 in ...
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Spring 2010 Exam 2 Solutions - Probability and Statistics...

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