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Fall 2010 Exam 2 Solutions - Probability and Statistics for...

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Unformatted text preview: Probability and Statistics for Engineering (ESE 326) Exam 2 October 26. 2010 This exam contains 10 multipieohoiee problems Worth two points each, five troe- false problems worth. one point each, seven short—answer problems worth one point each, and one freewresponse problem worth eight points. for an exam total of 40 points. Port 1. Moltigie—Choiee (two points each) Clearly fill in the oval on your answer card which. corresponds to the only correct response. 1. Returns are brought to a large deoartment store at at} average rate of six per hour. (This is a Poisson process.) If the store opens at 10:00 21,111., what is the probability that the first return will be brought in between 10:15 21.111. and 10:30 am? (A) .0498 egg/TAM PM @WW 3 (B) .0753 W 1 7:: 2e {fie/e Me B 3:33: Mi 3;; more) (F) 7769 ... / 44.5%) (I 45.55;} (6)7969 tire}; ""‘ be- (H) .8267 43/: 43 (1) .9247 j: 8 _.. 6‘; (.4) .9502 .-: , {7.33 2. Returns are brought to a large department store at an average rate of six per hour. (This is a Poisson process.) What is the probability that there will be more than 12 returns during the first two hours? w dose. 9.4... s... :i :2. 2 2 2:2 (B) .3113 . <62. 3185 ‘ “5:2“ :2 z» PE X :4 423 {(o) 434:1)? [DEX J (E) .4616 : )7 ... y S’r‘géfi (r) .5384 (”(3) moo :2: . #2453} (H) .6815 (1‘) .688? (3 ) .ssse La.) Let X be a continuous random variable wiih densiiy fix} 2 7:7, 5 S I ’ (.2 . X. (A)w a??? €3,512? (C) 62 (D) 62 w e (E) 63 —§— 6 1 (F) In :5 3: (G) in 5 (H) ln 2 (I) 133 w In 2 (J) in 3 + In 2 Find the median of The length of a healthy gall bladder is normally distributed with a mean. of 88. cm and 5: standard deviation of .5 cm. For What gall bladder length is ii true {hat 75% of healthy gall bladders are lenger than this? (A) 8.22 cm (8) 8.3? cm W MC) 8.46 cm § (D) 8.55 cm (E) 8.6/3 em (F) 8.96 cm (G) 9.05 cm (H) 9.14 cm { I.) 9 . 23 em (J) 9.38 cm {I’ivfifiérm (2-5”) 3133’} .§\) 3 Q L713?» 5. The life span (in months) of a eeriain starfish species is a Weibui} ranéom variable with (1» x .003 and ’3 a 2. What is the prebabiléty the): a ceriain membez' of this sgecies WU! five 3% to 2 years? WW .8589} (A) .9952 (B) .9153 (e) .9859 {(33) .2907}? (5) .2799 ('5) .7291 (G) .7993 (H) .9559 ('1) .984? (J) .9998 $5893 F528) :3- 25‘98’}: ~» 8589‘) . f. ‘ _. mmfiugfi} .1 89593589 5 if 8: 2 , 25353"? The following diagram shows the reiizzbility of each of the four (independent) components in a system. Find the refiability of the Whoie system. (A) .4815 (B) .9888 (C) .5112 (D) 5184 (1:) .7522 (G) .8589 (H) .8998 (I) .9858 (.1) .9998 919,81”; 8»? 99258“ me 8.9; 8.8958; 8/ i - 5 :5 Eiiééfé fig? 7". S'oppose (X, Y} is a two—dimensional continuous random variubée with the following density. fX3/{:E.y) : §x+3y {) < y <x <1 Set up the intengS) needed to find 1’ L X Y < 13 2} (A) f} 11% (321 +3 3y) dg; (£3; (E) Iii]: 5 (“Ex + 53;) 05;; d1: (5 :(3T+3y)dgdsc+ff i(3x+3y)d3d’rr 8. Let (X, Y) be a two-dimensional discrete random V23rla‘ole. The following information is known about (X, Y). ELK} : 2.6 FY3 33 3.4 E3X‘3 m 7.0 E[Y3§ n 13.5 1133/ :3 5.7 Find the correlation pr between X and Y. (A) m 9573 (B) A 553:3 (C) “ “H VX5355: :3 533323 3., (2,353?” s: 32335 (D) #3400 g _ 3 fi 2. (E) 43575 “3/1333? 3' E353“ 5“ 533“) 1“ , 33-335 (F) 30576 M 34' "G .1350 .3 W; , m 3} O [33333 ‘” WWW e "‘3 gé’fig (H) .4113: 3 \l {32.3%}5, 234%? (3) .5535 (.3) .9573 9. Let (X, Y} boa two—dimensional discrete random variable wirh density given by the ibllowing table. Find the average value ol’Y giver; a? x 3. fy:3 8 11 B? 2 i ,10 ‘G5 .05 3 §.oe .10 .15 .101 4 ‘. 05 e .mi {A)om5 (m ow «WWLWW« -MMMWWWWWMMWWWWWMMWWWM mlooe §;H’ mégé $§a§i rag §§7§ ya§ (o)3o giiwfi m)3e W,fi;§ W (m a9 gimfiffidzgg ‘6) “25 “ “ ' v {V 9; ~ 5 v "”3e523§{.2£3 WE, : {gjzgggsfi a, 2; law; «2* an 3?; fflllLb LU)“: (32»..339 T: :2 2g: (n we ’ 10. People often think of a randomized comparative experiment in the context of medical research, our the method is used successfully for research in many fields. The movie “Experimenzal Design” featured a socioiogical problem which was investigated using a randomized comparative experiment. What was this problem? (A) animal abuse {X (8) domestic violence J mm (C) divorce (D) drunk driving (E) gangs (F) high school rerention (G) racism (Pl) sieroid use (I) teen pregnancy (J) voter fraud Part 11.. TrueFalse (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is false l - .. . . K . 7‘ .I < .’ < l 11. Let X be a continuous random variable With denSity flay) x { 3 if 0 '''' l 5. G elsewhere 1 - . . . . . . . . T',‘ , ' < ., < -._ Then the cumuiative distribution function 13 FW) 3 { or if G W T m a. {l elsewhere as éfinm' 6:} i w . , W We}: any. Egafifixfifi . 8"; l2. Let X be a normal distribution. (X is exactly normal, not approximately normai.) Then the median of X equals the mean of X. ”g g: gg g .. Xfmw @fiauim W «fiery/wmeZ—«éiw 13. Suppose (XJ’) is a two~diniensional random variable with 1(ng 15, LY| :4. and EEXY] : 60. Then X and Y are independent 52 j; X W M nave... WW gig/mag new ggXVE-i gffijngj ii [gnaw “fie, ommm a}: fingé own... 14. Let (X,Y) beatwo—dimensionalrandom variable. Then —l g Cov(X,.‘/)< M W flag—43% V? f ffix‘ég f 5} wife woewazieflee a“? X ' .. if v; i" W V We give. new jean?) eueavgé‘ee . NJ 15. The curve of regression of Y on X always comes out to be a linear function. a ; WM J41 7% m Maj! new“? 3% WE Part Eli. Short Answer (one point each) The answer to each. of these is right or wrong: no work is required. and no partial credit will be given. Give only one answer to each. (If you give more than one answer. the poorer one wili count.) l 15. Let X be a continuous random variable with density flea} 2 w '1 g :r < .9. Find the exact value of 139;“ f, i I 53; 55X; «3 Mg: Edi; 3 x g a u: f: M E; l7. Find the exact value. of H18). . t , l ' a , 1‘ w . $555332”? 3 see gee”; l8. Let X be a gamma random variable with o; x 5 and 13 z 2. Find VsrX. ! g3 1 ’32; a tree“ :35 5;" sari/3% :— 21$ l9. Below is the graph of a failure density. Let t he a fixed positive value, as shown. Indicate exactly where Pitt) is in the ictnre, where R represents the reliability function. Your answer must he completely clear in order for you to receive credit for this point. l l l § e 20. Fill in the blank with the correct one word answer: A(n)w is a random variable whose numerical value can be computed from a sample (Note that ‘estimator” is a closely~related word which as not the intended answer here.) 21. Fill in the blank with the correct two—word answer: Let X be a random variable. A(n) g g =59;- M g is a collection of independent random variables, each having the same distribution as X. 22. A sample of size n z 5 is taken from a large population, resulting in the following date. ll .15 28 £6 20 Find the exsct value of the sample variance. Hint i: if 2 18. "Hint 2: Since work does not need to be shown on shortenswer problems, you mey do this by hand or using your celculator. Par: EV. Free Resgoase (eigh‘i poinés) Follow directions carefully, arid show all the steps needed to arrive at your soluiion. 23. Let f X , Y} be a Ewe—dimensional coatinuous random. variable with the following 305m density. ) fxv'iwlziwe’f 5>0 53:3 ., 2 U (a) Find {he marginal density ffix) for X. Make sure your answer includes not only a formula but also the rage of values to which the formuia applies. {75 , g 3 y (a 5 2 j a. 3 Jr :3 .5.-. riled! - M 5. ”:2. .{al} .5 gage. 5.. ~55 efigfé 5:5; ‘ E i i' m _. §2€"x2/§”53 3 Eaefi“x fled/:3 (b) Find the marginal. density jig/(3;) For Y. Make sure your answer includes not only a formula but also the range of values to which the formula applies. 57/»: 555;” J. .52.. -5: W3? “if; 221?: '75 ”as... ong& ’96 .. f i g 73“ " 5 i 5 '“ g” 5 .. 2" 5' ”1 é; «a ea .27 g, 5 C? .5 i . . . 35 “7"” E fl 5555’ f: (c) Find the conditional density fy'lw for Y given :5 Simplify as appropriate. Make sure your answer includes not only a formula but also Elie range of values to which the formula applies. . 53- ' - 5 f4 _ .. 1 l3 i.— 55 W .. 532...; x 5 all}: ’ .. r“ WW... :5» we! if g. 535;} 5’}: aim”; if 557; :gegj‘. $.ng {d} Are X and Y independent? Give a reason to support your answer. Your reason may be orief, bu? make sure it is expressed very clearly. ‘" l .5 5 - o 5 ‘5 .5? 5‘5 gee. $55.52-.5W5af onaejjjm jlxéx;*;..1. {EV i”; - El {1‘} f 5:3 r ...
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