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Spring 2011 Exam 3 Solutions - Probability and Statistics...

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Probability and Statistics for Engineering (ESE 326) Exam 3 April14,2011 This exam contains eight multiple-choice problems worth two points each, five true-false problems worth one point each, eight short-answer problems worth one point each, and two free-response problems worth 11 points altogether, for an exam total of 40 points. Part I. Multiple-Choice (two points each) Clearly fill in the oval on your answer card which corresponds to the only correct response. 1. Arandomvariable Xhasthe followingdensityfunctionwithgunknown: f ("): g2re*0', r) 0. Let X1, ... , X, be a random sample of size n from the distribution of X. Find the maximum likelihood estimator for 9. \ -/ n (D) + (E) + (F) # (c) # (H) # fj * cll.' \ +-f:rI€" ) :Z_ 2. SupposeXisanormalrandomvariablewithp,:50and o:3. LetX1,...,Xa bearandom sample of size n : 6 from the distribution of X. Find P[X > f t]. (A) .0228 J,*ntu-Lro, 4 X , f--\x I '7 ftel .zozil f) r *4- r (c) .zbzs ''tns>'"ta-t , .A; : {d' ' 5}' = ttr (D) '36e4 rLorv4ul.-t4: ({' , lac' , {c %;\ = ' za 1 t (E) .6306 (F) .7475 - 0f - (G) .7929 ^ r -- -? . (H) .sTT2 VLx'tt J= PLz : 2A'7 I
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3. Let X andY be independent random variables with a; : 5 and oy :2. Find o2x-y. I VorY = L{ VonV = * (A) (B) (c) (D) (E) (F) (G) (H) (r) 8 t2 18 22 46 54 t/46 JA Je6 i*n bx- ?) = i^. ( t' x'r Gt) Y ) 2l = 2 Vo..X '"t- /\r\ = ta+ (:,)'ln4 (,>0t) = | ,-751 (J) ho4 4. Let X be a random variable, and suppose [.L1,,[2] is a confidence interval for the mean p, of X, where c : .05. Which of the following is an appropriate statement to make about this interval? (B) We are 95% confident that p lies outside this interval. (C) We are 95% confident that z lies in this interval. (D) We are 95% confident that r lies outside this interval. (E) We are 95% confident that X lies in this interval. (F) We are 95% confident that X nes outside this interval. 5. Suppose a random variable X is known to be normal with standard deviation o :12. sample size is required to estimate trr to within 2 units of its true value with92%o confidence? (D 110 / 'Yt*>- ( 'K',-(+Y Z* = i,tr,'P," '* (,qt) (,.tt') ( 'r\ ) = lt0,+
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