Spring 2010 Exam 1 Solutions

Spring 2010 Exam 1 Solutions - Probability and Statistics...

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Unformatted text preview: Probability and Statistics for Engineering (ESE 326} Exam t February 18, 2010 This exam contains 11 multiple-choice problems worth two points each, six tme~false problems worth one point each, seven shott~answer problems worth one point each, and one free-response problem worth five points, for an exam total" of 40 poiats. Part I. Multiple»Choice (two points each) Clearly fill in the oval on your aaswer card which corresponds to the only correct response. 3. Each time Acme Supplies lands a new customer, an eight—character alphanumeric code for that customer is generated randomly by a computer, Each character in such a password can be chosen from among the 26 letters AWZ and the 30 digits 0—9. Repetition is allowed. What is the probability that a randomly-generated password of this type contatns all letters and no numbers? (A) .0005 My ’ a (B) .0200 4&9 (C) .0223 3 [A ‘3 (o) .0386 (a) .0516 f (a) .0700 l (G) .1305 (H) .1712 (I) .2523 I . wife 2. What is the probability of getting a “straight flush” in a five-card hand dealt from a standard deck of 52 cards? (A straight flush ts five cards in a row, all ofthe same suit, such as 31', 4v, 5", 61!, 7?. film. An Ace can be low or high. In other words, both the straight A, 2, 3,4, 5 and the straight it}, 3, Q, K, A would be allowed. (A) 0000002 (B) 0000139 * f5 ((6) 0000154 . {a (1)) 0000300 ( _ (E) 0004790 (F) 0004052 ((3) 0019054 (H) 0030240 (I) 0030400 (J) 0504230 3. Bekah, Kendail, and Mary go shopping for shoes. At Fortuimus Footwear) there are 250 styles of ladies‘ shoes. Each teen pians to buy either one pair or two different pairs today. (It‘s OK if muftiple teens choose the same styie.) How many outcomes are possible? (A) (250)3(25002)3 MA. 5121A we“ we pew (B) (250)3(250PQF (c) (3>(250)3(25002)3 a (B) (3)(250)3(250P2)3 W m" 5‘5? 5"” PM“ ( (a) 3) 250 C: ' " < [ + 250 3] [25/536 4 air; [25:23 a? 25361:][2,%+ $32152] (F) (3) [250 e 250%] Héwajafi [wage flu is}; gig/sis 19 $2 M (G) [250 + 25002] 3/}; 3 (a) [250 + 250132] (I) (250)3 + (25002,)3 (D (250}3 + (250P2)3 4. 8% of Americans have diabetes, 12% have chronic kidney disease (CKD), and 9% have CKD but not diabetes. What is the probability that a randomfly~seiected American with diabetes also has cm)? M {4: wake. WWW Lam 83 32h, {22/9/1165}: Aka CH?) PIA] :: ,5)? (9537 3 J2 Plfi’sgj at}? (A) (B) (C) (D) 4 m3; 4 2 (3)5 32 5 3 EH $01M OOH-4 5. 15% of Americans over 30 years of age have earned an advanced eeiiege degree (anything beyonci a bachelor‘s degree). Of Americans over 30 with advanced degrees, 22% earn over $100, 600 per year. Of Americans over 30 without advanced degrees, 6% earn over $100, 008 per year. If an American over 30 earns over $180,003 per year, what is the probabiiiiy that he or she has an advanced degree? F} 3’ WLL gee/wen him we QC/féflrflgzl r *- " ‘ .s (A) .0459 (B) .0481 i3: ‘f’i—L fog/WW1 eye: .13” {607343566} fem (0-2??? Pfajriig‘ ‘Pfeiflxin Pi’xele‘lriw .3529 Lida ~ ; mines-awe: a“) ‘69” » aim/m (G) em PINE] == - 7 2:. “392% (H) .7273 (12258355) + name) (I) .9519 (J) .9541 6. Consider the foiiowing function, which is potentially a density function for a discrete random variable X. f(x)=f; :1:=2,3,4,... What would the value of the constant c have to be in order to make f a density? (A) £5 (B) e (C) (D) (E) (F) (G) 3 (H 4 .. whfi n3ha Mir—é via-3w w CZiZ (5) £6 )- 7. The number of “hits” to a certain website is a Poisson process with an average of 5 hits per hour. What is the probability that the website will receive exactly 14 hits in a two—hour period? {(A0 .o521 ) (B) .0663 ((3 .0835 GD) .1157 (s) .8243 (F) .9165 (o) .9337 (H) .9479 Newt Isaaos is somewhat prepared for his physics finai exam, which will consist of sixty questions. The probability that he will answer any given question correctly is .75. What is the probability that Newt will answer between 45 and 50 (including 45 and 50) questions correctly? meWe 23:5" sxe keno P[X=r4] “3 goat’s: (A) .4054 (B) .4294 M“,ng =71 : 4,3 F. '3: 3 “75” (C) .4524 £ 4’ l n # ‘ H. 4764 P [45% X s 5e] : PIX :e is] w P1X 5 w] .5235 - ISéts .5706 .5946 “r .9§4§3 M Aim-3o :2 .52348 (G) (H) There are 1200 booby birds on Espafiola, (one of the Galapogos Islands), of which 750 are blue— footed boobies and 450 are masked boobies. (Y es, these are real species!) If a naturalist captures 10 booby birds at random points around the island for study purposes, what is the probability that at least eight will be blue—footed? (A) .01 Nf 4”: 12536} AW: 752} is. :16 (B) '06 Pfx'as] ePfX‘fije Plsegjepfx‘fiioj (C) .15 (gaze???) $9 (ejéj/ifi) “if {ier/ifej as) .79” :: “mesmsmm.owwmsss F , 51m 5: zoo g ‘3 25m: 23)) K m {l m ) (/ so ) (H)sg ::.J4op«eiosfio~esoo2@ s..soee 10. A veiieybail coach hoids a serving competition for her team members. Each girl makes repeated attempts to serve the bali over the net and in bounds (a good serve). When she has reacheé a total of five poor serves, her turn is over, and the tot‘ai number if attempts (good and poor) is counted as her score. The girl with the highest score Wins. Ionian can manage a good serve 72% of the time, What is the probability that her score will be 13‘? (A) .0036 Lem?me JL- ‘2: 5 f3 ‘-= i 2.? (B) .0094 ‘i , w, ,. . . ((0) 3615? P1X": f3] '3: (if)(/:Z§§S(?Z>g (2)) .0984. (E) .1690 a: ems? (3?) .8460 (G) .9016 (H) .9385 (1) .9906 (I) .9964 11‘ Let X be a hypergeometfic random variabie with N = 24, r = 20, and n == 8. What are the possibie vaiues for 32‘? (A) 0,1,...,4 (B) 0,1,...,8 (C) e,1,...,20 (D) 1,2,...,4 (a) 1,2,,..,s (F) 1,2,...,2e é 5: ’2 “if g (G) 4,5,...,8 i x‘ :31 26:3 (H) 4,5,...,29 gay: 5; a; fl :5 1:: 41 (I) s,9,...,29 ‘ ( 5, ‘X m ME mews (I) 8,9,...,24 Part I}. True-False (one point each) Mark “A” on your answer card ifthe statement is true; mark “B” if it is false. 12. If A and B are events such that P m 0, then A and B are mutually exclusive. “flake: Eli-{rm i3 f 1?" {53% 53} x @313}: a g 13. Suppose A and B are events such that PM} = .'?5, PER] : .6, and PM U B] m .9. Then A and B are independent. , _ , . . flame} em} a eiel « eiaeel ghee Fiaoeg he???” :1 eia‘éwie} 14. If X is the amount of time that goes by until a light bulb burns out, then X is a discrete random variable. ‘ , , f; E l: t if; W36 X gem aaaewm W we ewe Maia.wa «3:; .. i? gm ; g g i .. y “*5” amvh-é‘ee} ae we a}: em. oboe, fiugwiuzaLé§ . 15. Suppose X is a discrete random variable with E[X] 7» 12 and 0X 3 3. Then, according to Chebyshev's Theorem, H3 < X < 21] = g. p aegipfixijv viz: {/éisag-eufwaewfe “Wan {fivfiu‘i ea ewe; ééséw ,, a a l on r. 1’} é? 5“” five 5 «w ammo 2 - 2 16. Let X be a discrete random variable. Then EFX2 + X x EFX + E EX . l E i ; a; If”? ’” .3 flag" g 2 axis} fix l a» :sz fax-E} a. go}? fee 17. The St. Louis Rams play 16 games during the regular National Football League season. Let X be the number of games won by the Rams timing the season. (There are no ties, only wins anti losses.) Then X is a binomial random variable. i {x 'W' F 3’ °' 9. l. f V I' r3 5.} f M or“ '5 ’13; ace-:2)... Sign) jean»??ng ; 'gffi' 4’? 7w If; ‘5 i it 9/ ea. Vie-e5 a. eemegaaoe , Part II}. Short Answer (one point each) The answer to each oftbese is right or wrong: no work is required, and no partial credit will be given. 18. Ellie is one of eight swimmers competing in the 200 yard butterfly event. Why would it not be appropriate to say that Ellie‘s probability of winning the event is %? (The reason can be stated simply and briefly. If you don‘t know, don’t make something up. Your answer should be precise and should use a technical term.) “Wee ceeefi-efiwwa, ewe; aWW? 5/ edge fie 4 <5" ; J 1 w a V ' jWflgf}Myfi§}t 3 i ales ride?!» as we. 57/ "ma/Limes. a, 5 = 2 g , (6" ,J m ewggiefizzasm‘“ ieeeéwaazg u. 19. Suppose that A and B are independent events such that PM] 2 .55 and P[B§ m .72. Find P {A113}. 5’ E e 5e] 2 e [is] a: ,5? eia Wise — mead; For problems 20 and 21, suppose X is a discrete random variable with Em 2 12 and 0X K 3' 20. Find BEX + 8], gZ—éx “g” g}; éélfix] e E? :3 i4 21. Find Var(%X+8)‘ lair "‘5 idem/3(- : :3 For problems 22 and 23, suppose X is a discrete random variable with the following density table. m 1 a s 9 fix) .3 .2 .4 .1 22. Find E‘le a :3 {£363} elf/side) +Z§lé¢é§ ec/e'ldii :2“ :3”? a 23. F ill in the density table for the random variable X2. m s 2:; ed 3;; 9W s2 A? is We 24. W113: is the variance of a negative binomiai random variabie with r m 10 and p 2 .4? Waite your answer in decimai farm. _ gm 6 {Eeyex‘} M m awry r w w {3375’ Part IV. Free Response (5 points) Foilow directions carefully, and Show ail the steps needed to arrive at your solution 25. The feflowing function is a moment generating function fer a random variable X. mX(t) fl .00032(2 + 36t)5 (3) Use the moment generating fimctian to find ELX 51% 3 Wig :1?) : eeeegz 557%? e 36%} {Egg i , {Ziéfi’égfii'g (’2 «5» 3g? 5: Effie” £3’foéé’EZ jéw ‘3‘ ,zféeé??(é>{4§éfi> 3 3 (b) Use the moment generating function to find Var(X 2 . A 2% 59 3» deeégfifi 55%;)524» 3e‘é535’3efi e éeeéziggéYNEe/éfi d9}: 1” ., &§?ai e‘E£/Z+ 36$? 4 geeéigefifg; 3512}? “Z r”? “ “WI: 6 ‘ r = 51%;?" M2» 43%)}. 3 ewééiaéfib meawwgfi} e25 :1 £5332 VWX I 2&2“?! : $32; ...
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Spring 2010 Exam 1 Solutions - Probability and Statistics...

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