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Fall 2010 Exam 2 - Probability anti Statistics for...

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Unformatted text preview: Probability anti Statistics for Engineering (ESE 326) Exam 2 October 26, 2010 This exam contains 10 multiple—choice pro‘oiems worth two points each, five true-false problems worth one point each, seven short—answer problems worth one point each, and em: free—response problem worth eight points, for an exam total of 40 points. Part I. Muttiple-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 department store at an average rate of six per hour. (This is a Poisson process.) if the store Opens at 30:00 am, what is the probabitity that the first return will be brought in between 10:15 am. and 10:30 am? (A) .9493 (a) .0753 (C3 .1733 9)) .2031 (E) .2231 (F) .rres (ca .7969 (en .8267 (o .9247 (3) .9502 '2. Returns are brought to a iarge 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? LA) .risa (a) .3113 (CD .3285 93) .4240 (s) .4616 (F) .5384 (to .5?69 (to .6915 (n .9997 a) .8856 LI.) Let X be a continuous random variable with densii‘y f(:;1'7} : i e < :2: g (:3. Find the median of .1" A X. (A) 615;? (B) 83;: (C) e2 (D) (22 m (3 (B) 62 w e: (E3) mu} (G) in? (H) 1:12 (1) ln3-ln2 (J) in3+1n2 The length of a healthy gall bladder is normally distribuied with a mean of 8.8 cm and a standard deviation of :5 cm. For what gall bladder length is it true tha£ 75% of healthy gall bladders are longer thaw this? (A) 8.22 cm (B) 8.37 cm (C) 8.41.6031 (D) 8. (E) 8 (F) S. (G) 9.05 cm (H) 9.14 cm (I) 9.23 cm (I) 9.38 cm The life span (in months) of a cez‘tam starfish species is a Weibuli random variable with n: 2 .003 and fl 2 2. What is the probability that a certain member of this species will live 2% te 2 years? 55) .9052 (B) .9153 (CD .9350 GD) .2007 (5) .2799 (F) .7291 (CD .7998 61) .9550 (0 .9847 (J) .9948 The following diagram shows the reliability of each of the four (independent) components in a system. Find the reliability of the whole system. 55) .4815 (B) .4888 (CU .5122 0)) .5184 (E) .?128 (F) .?522 (CD .8580 (99 .8994 (n .9855 (J) .9998 Sapposo (X; Y) is a two~dimonsiona§ continuous random variable wk}: the: following donsiiy. hwWfl‘x§[email protected] OSyS$Sl Sei up the iniografls) needed to find P E X - Y < E 4 (A) flfjj$i(%x—+3y)dydm '3‘ (B) 1;; f 33: + By) (3y do: Let (X, Y) be a two—dimensional discrete random variabie. The foliowing information is known about (X , Y). flXhflfi fmfiza4 Egflmzo EW¥xnfi .fiXflxSY Find the correlation pr between X and Y. (A) "29573 (B) va5833 (C) wa4114 (o) waléOO (E) -—.0576 (F) .0576 ((9 .1400 (rm .éllé (o ‘5833 (3) .9573 Let (X, Y) be a two-dimensional discrete random variable with density given by the I‘D-Rowing iabfe. Find the average vaiue of Y given 1‘ x 3. 23 ‘ 8 I} 3??fo E 3 (F) 4.9 (G) 11.125 (H) 11.25 (I) 12.25 (3) 12.5 People often think of a randomized comparative experiment in the context of medical research, but the method is used successfu’ily for research in many fieids. The movie “Experimentai Design” featured a sociologicaf problem which was investigated using a randomized comparative experiment. What was this problem? (A) animaiabuse (B) domestic "violence (C) divorce (D) drunk driving {8) gangs (F) high schooi reteniion (G) racism (H) steroid use (I) teen pregnancy (j) voter fraud Part 11.. TreewFaise (one pointeaeh) Mark “A” on your answer card it" the statement is true; mark “B" if it is faise. .5 O elsewhere i _ ‘ . . . . . . .. :.~ , f } < ., < 2 Then the etimulative distribution function is Fm} 2: { ~33" E E W T r“ ). 1. ~ «- s . . v . , ~ .— ii < f,” < ii. Let X be aeootiiiuous random variabie With densfiy fizz} :x { U “‘ I — 3. 0 eisewliere 12. Let X be a normal distribution. (X is exactly normai, not approximately normal.) Then. the median of X equals the mean of X. ., 1.3. Suppose {X,Y) is a two—dimensional random variable with EA” : 15, ElK :4, and E[XY] : 60. Then X and Y are independent. 14. Let (X: Y) be a two—dimensional random variable. Then m1 3 Cox/’(‘IXfi Y} E 1. 15. The curve of regression of Y on X aiways comes out to be a linear function. Part 111. Short Answer (one point each) The answer to each of these is right or wrong: no work is required, and no partiai credit will be given. Give only one answer to each. (If you give more than one answer, the poorer one will count.) 16. Let X be a. continuous random variable with density flat} x i 1 3 r S 6. Find the exact value of {Xi “ 17‘ 18. £9. 20. Find the exact value MIX-10). Let X be a gamma random variable with o: z: 5 and ,3 r: 2. Find Var X. Below is the graph of a failure density. Let t be a fixed positive value, as shown. indicate exactly where Bit.) is in the picture, where R represents the reliability function. Your answer must be completely clear in order for you to receive credit for this point. Fill. in the blank with the correct one—word. answer: A(n) is a random variable whose numerical vaiue can. be computed from a sample. (Note that “estimator“ is a elOsely—related word which is not the intended answer here.) Fill in the blank with the correct twonword answer: Let X be a random variable. Mn) is a coilection of independent random variabtes, each having the same distribution as X . A sample of size n 2: 5 is taken from a large population, resulting in the following data. 11 15 28 16 20 Find the exact value of the sainpie variance. Hint l: X 2: t8 Hint 2: Since work does not need to be shown on. short—answer problems, you may do this by hand or using your calculator. Part W. Free Response (eight poinis) Follow directions carefully, and show all the steps needed to arrive at your solution. 23. Let (X: Y} be a two»dimensional continuous random. variable with the following joint densiiy. er'lxyI/l x 5395‘”; 1‘- > 0 1 < y < 3 (a) Find the marginal density f fix) for X. Make sure your answer includes not only a formula but also the range of values to which the formula applies. (b) Find the marginal density fy(y) for Y. Make sure your answer includes not only a formula but also the range of values to which the formula applies. (c) Find the conditional density fygj, for Y given r. Simplify as appropriate. Make sure your answer includes not only a formula but also the range of values to which the formula applies. (d) Are X and Y independent? Give a reason to support your answer. Your reason may be brief, but make sure it is expressed very clearly. ...
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