Spring 2011 Exam 1 Solutions - fiebability arid Statistics...

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Unformatted text preview: fiebability arid Statistics for Engineering {ESE 326‘} Exam 1 Febmy 1?, 291‘: Thie exam centains nine muitiple—eheiee preblemg weft}: {we paints each, iha‘ee truefame problems wefih ene gem: each, 18 shofiwanewer prebiems werth One pain? each, and {we free-respen3e prohEems worth nine points aitogether, for an exam tetai 0f 40 points. Part I. Muitigie-Cheiee (two points each) Cieariy fiii in the med on your answer card which corresponds to the 0:11}! corms: response. E. Each time Acme Supplies Iancis a new custemer, an eightmeharaeter alphavnumeric code for that customer is generated randomiy by a computer‘ Each character in such a passworé can be chosen from among the 26 Eetters AwZ and the £0 digits GL4; What is the probabiiity that a randemiy— generated pagswerd of this type contains no repeated ietters er numbers? (A) .0000}: (E) 390025 Qgfiej <3“: I (a) .2523 W" ”“3“” 1" an? «£53 25*” {D} .3013 3 519 e 3&5 (F) .56?5 (G) we; (H) m7? {1) .999975 (1) .999989 so In how many ways can six peeple be chosen from a pee} ef seven femaies and nine maies, where at ieast one female and one male must be inciuded? {e354 NM 5:: f‘eé fl (3391 {jefiwié/J“ géjg-i: "93%? (C) 189 {:3} 882 (E: 19?:- {F} 1268 »WMwa—M [M ““x 5 {H} 8008 {i} 63? 85:3 {3‘} 1; 5135512 3. What is the prokabiiiiy that a sevenucard hand dram {mm a siandard deck 43f 52 cards centains (me three‘wofwaukind 311$ Wm (éifferentj pairg? {Am examp'ie {:sf 3 hand like this weuki be 5¢,§Q,Sv,Qv,Q¢,2é32v.) {1763: you peker afic-ianados, I mnéefsiand-that {he secs-x75: paia‘ is irrelavzmt singe- {mly five 9)? ihe: cards; cam be: ufiéizafi, but Earner me 311% campuie the pmbabiifiiy amway.) (A) .0083 1/ f3 jib; Effgay ‘1? g]; £912 < fifigfigagfilf » KB} 709?”: WW MWWWW : f flfifi? @ f§3> (D) .8012 g M; (E) .0018. (F) .0024 (G) .00?4 (H) .0210 (I) .0222 (I) .0443 4- 28% of American heusehclds own a dog but not a cat, 15% own a cat but not a dog, and 51% awn either a cat er a dog er bath. What percent ofAmerican househokds mm both a dog and a cat? ex) 7?3% m a {£3 12$§ $3) :3?6 (E) 16%3 (E3 20%% «3} 21%3 @&) 34%e g Q) éififi {3} agag LI: Suppose A and B are events such {hat PE 2: 4 Pig/1H8} 2: .75., anéi Pffiééj m .6. Find. PiB! [A .096 "E- f” y r,“ f r} fi A ”E _/ {”32 M? “5‘ 1' r( . 1‘ m. 5 _ H fig??? :5 :1 f} 5:753»; ,3 : 953%; w {Veggiig} w. £23; {8} m4 _ {C} .18 :32 gag-$52] :in’f’zgffigéfiéfij (D) .24 (B) .3 WWW A g ”E w w w . 5(5) .32; g“; LEE“ * WM ‘3: “£2“ MWJ at M? 5, g (G) .45 (H) .5333 (I) .6??? (I) .8 b»: Consider the population Of American maies ages 18 to 65. 70% if those who are 18 to 39 years old (“younger guys”) are empioyeck, 92% of those whe are 31 to 50 years old (“middle—aged guys”) are empioyeé, am? 86% of those who are SE :0 65 years old (“oider guys”) are empioyed. Withifi this whoie popuiation, 25% are younger guys, 60% are midéfiaaged guys, and 15% are aid-er guys. if an American maie 18 to 65 years old is employed, what is the probability that he is an 016$; guy? wmh. “M,D,W : Q} T1507; 53/] : gig/«:m—yg/bfi. 43% M {/5 g" a '4 5 (B) 1578 {#2:} 9 M ”{Ejgwfgfi Hggfiivj K‘gM afifiifiw “Mgr/y: (C) $655 g i ”W " <3; (D3 .31? {E} .1789 (F) .1950 y. w? W M (G) 2044 Pi €53 “ 25" a s» Mg m : {H} .2632 5? E, 2;; i » 2 M1 {1) ‘355: :3 35:} g, T £5: {1) .6449 * “ ~45 5 - ”E {,2 322$ 1235’: Rafi P g {in} i g 1! “A: WWW ‘3:me :‘M “2“": g“ x 5327535233”; ; 5%3;&&6> «4 5’ a: £5” E; if: 53/3”? Grandpa Den heiiis an egg hunt fer his granfiehiiéren every Easter. Re puts a quarter in eaeh of 40 eggs and gets a defiar hilt in each 0f eight egge. if Kim finds nine eggs this year. What is the pmbabfiity that she gets exactiy twe 9f the eggs with (tenet bills? . .s‘w * f M W 81"“ M ’1 {A} ’00?1 iii.i%§g£f%m§é’%% .53 .. gig”? we .. § we. 1 % ix".— ,- {B} .0988 f 3 . g . . es. . {C} .911: §>§XfZE : {:Zémij .. “$3 (D) .054: .‘ggxz .. = (E) .1359 { (F‘) .2472 (Gt .391 5"” M (I) .3421? (I) .3662 Cowboy Bob flips a coin every {fay to determine whether or not he Wiil take a bath that day. If he gets heaés. he takes a bath; if he gets taiis, he (toes not. The problem is that he uses a weighted coin (given to him by his gmnddaddy, Gambier Gus} which comes up heads oniy 40% of the time. if Cowboy Bab censiders Sunday to be the first day of the week, What is the probability that his first bath 0f the week will be on Tuesday er Wednesday? (A) .6864 Q. #@%g > & g 5 (B) .1344 55% f 2 . . 2. . f). 1:432. P51? :23 .x. Xe] : 5.5.} {L9} é» 515.3 5%} (D) .2304; (E) .3841} 7: . 2.353%; {F} .6168 (G) .?696 {H} .8569 (1‘; .8656 a) .9136 ‘9‘. Than: are 2% studentg emailed in an imméucmry” anthrapgiggy mama Gf theme, 203 are: acimail‘y in tha (£333? and five are 011 the waifligi. Suppase there is a 2% shame that 2111}? given student ammg {hit 209 WEE withéraw earl}; emugi‘: in the Semestar It; aliew SfifilEOfiE fmm the waitiist Se be admitéeii inw the Class‘ {Yen may asgmne thai méihdmwais an? indepenfient 0? {me anoihaz‘. and yen may aiso assume that .119 one from the waifiist mithcimwg.) What is the probabiiity that 33 the students; on the wafifist wiif get Emit) the class? {m,m% éggm®@g fizfifififi §>$5£E (B) gym M (C) 2815 PEXEESE EEW'PEXEEEE L“? W] W . __ (E) .4886 “‘ aéZéfi‘aw wflng (F) ‘5994 ((3 ,6288 (ED .?185 a) .?867 (I) .8914 Part II. True»False (one point each) Mark “A” on your answer car& if the statement is true; mark “B” if it is faise‘ 16. For any events A and B PEAEB’EM M 1 w PEAEBE. - m 3-z3m {iv/WM ‘Wéfijg £5,3ng if (Lazy? {##3ng if; @1me E R if .7 , ‘ , W “ .533, M“; j E3 “Em/x ’Ev’iafi éfiyuxm {'13 a, 5/: “Egg gsgfiéflixzamfli’ig ,-' if ikmi iii 1%.. Let F be the camuiative distribution function for a discrete raniiom variabie A. If x} g x3, than gram £44,, flu, a.g‘&..§«§g7é ff 8 , sé ’? f E ”’i 2 fl 3%. “Law. 5:“ fiaawwwggaga; fE/wig‘zaéwgé}, E mi M WM at flé’fifiéfiyhgmgff? fiéééfl,£;2~i_.§’}€ ‘ i“ 12. Let X 3:36 a discrete- ranéom variabie. Thar: fer any reai amber}: c, an); m sax, faww vagjsé’ w gfigfl; Pan? HI. Sheri answer {one pram: each}. The answar ta each 0? these is right car wrang: m3 wmri: is required, and mi) partiai credi? WEE} be given‘ Géve (311%); one 531332133" is eaeh, (If you- gisse mare than em answer, me pasta? one wiil Gaunt} Far any pmbiem requiring a numeric-3% answer, give a numaricai answer, net 31:31: a femuia 03' an unfinished camputafim. i3. What": singie afijecfiive dascflbes a phenamenon fer which may individual remit is unpredictahle, bui a pattern emerges in the tang run? Efiawmygwg 1.4. and 15. Let A and B be events such that PEAE m .3 and FEB] 2: .5. Exactly two of the foilowing Statements about A mid B are true. Which ones are they? (I) If A and B are mutualiy exciusive, then P EA U B} a: 8. 3/! (II) If A and B are mutuaily exclusive, then PEA fl BE 2 .8. (HI) If A and B are mutually exciusive, than PM U BE m .13. (IV) If A mad B are mutuafly exciusive, filer} PEA 5‘? BE 3 .15. (V) if A and B are independent, than PEA U B; r: .8. (VI) If A and B are independent, then PEA {”3 BE 2 .8. (Vii) IfA and B are independent, than PEA U BE m .15. _ (VIII) if A and B are independent, than PEA 5“: E 2 .15. w" Fer probiems 16 and 1?, suppose X is a discrete randem variable with ,u: a 50 and 0’ a: . i6. Fina Van-(3X + 4), 3 e. , £3. W ffil‘; 1- M fink-g- ,y’k W ”W, EM {3% ME} ,. fiixw-X w {Mafia} w fiéa 1?. Fifiin the biank: ff“ 2% W Awarding to Chebyshezz‘s1nequaiiiy,PE42 < 55? < 58E _ W E f‘w W W ,2: é; 133 19K 20. 21. Below is the density fer a binomial tandem variabie with n 3 iii} and p 2' .3, except that the values to which the female applies are missing. Fifi them in. f<$}e(%8)(.3)“‘(-73n“$ s: e g) 2} 9 ea} 3 Given a Bernoulli trial with constant probabiiity of success 39 2:: (16, what is the average number of triais needed to obtain {we successes? RE ".5“: g: * £3“ a“: «v- ('2 '3 - i; ?”e£»;.§eimffi,- awesmeeefi “’3‘; ”I '2 3””! g Egg s: M {W W #3”; g: g: X e W '3: g3. *5 g g e ’ " Suppose Ehere are 60 marbles in a bag, of which 25 are biue. What is the average number of biue marbies in a sample of size 18 taken from the bag? £133; .ma;_,,uwmfz:m 33,} répifi Me: if ”3’5; 3’ Egg if j K? 333:3 5/; 0,. w W: W M iigj” ngsege :23 Suppose X is a hypergeometrie random variable with 3‘5 x 2006, r x 53, and n. :2 2%. Since 3? is Earge relative :o m the hinomial distribution may be used to approximate the hypergeometric distribution. What value of p sheuid be used? 52:} g ”k j: W w” ., a“, f .4W 1W, Ezfiée; ‘9 5:)“ 35:; 93:13}, E‘me fie-mange (paint vaiues as shemm} FGEIQW directians carefully, and Shaw ali the Sifi§33 needeé {a arrive 31: yam saiuiien. If you Wish {0 mum any gamiraers, round a} fair decimai giacea. {6) 23. Cansicier “aha discrete martian: vafiabfie X Whese density is given by the faiiawing tabi’e. g i 1 2 4 8 :fi fix); .g ‘ .ag mg .OQGQ 30009 {3) Find E {X}. jg»; {$33555 2%K/Zgfla}%§g§§§& g;%/,§}Z/sg:é W!¥§€_£d&flu ”4%,me 5,; :3 y 5? we: :: 52;;5; {:3 :; g g I a; fi i“? if») > 1' ma. . m “:m‘dw“ h, I i .auo ”‘99 {13) Find waxy ngw'yyagayiévéfix gm 5;?» 1" g Q; x21 “—1 (:1; :E 5/; a} a»? «:2; ”a 3?:Mwwiwm ““ng M W gm; 322 y i; AM £53" ‘3“ 2 W Van3 :g’wass‘zfi’E *" 2J¥3?3 "P 2%; (c) Find ax, ’3‘; fi’xég‘wgw‘ :: ”95%; {3} 24. Find flue derivative 0f mfiflm A“ 'Eie’iV with respect t0 i. Yen do mt need to simpiify’ your answer: war :10 yam need t0 ping m anfihmg far i. ...
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