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midterm2solution - AMS 310: Survey of Probability and...

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Unformatted text preview: AMS 310: Survey of Probability and Statistics MIDTERM EXAM II Fall 2009 Last Name: W First Name: _______ ID: Show all your work for full credit 1.(12pts) For some constant c, the random variable X has the probability density function Km)“ (12:4 if0<m<2, _ 0 otherwise. Fdeégf‘E‘ZdLGriil-g> S41Crxti-M;,fl,7 og/gfia -? 6:}? gift: = Em; 1W” ~33: 4311;: 5% UMCX): “$3 “ (“é-“2.: 2. The time X (in hours) required to repair a machine is an exponential r.v. with parameter A = %. Find (at) (613135) the probability that a repair time exceeds 2 hours? mil-'l' I 2; I P(X>l)= e = e (b) (6pts) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours? MLa 7iOIX7‘?): P(X>f) : e“; 3.(10pts) A randomly chosen IQ test taker obtains a score that is normally distributed with mean 100 and standard deviation 15. What is the probability that the score of such a person is between 90 and 110. pwosxeno :- H‘lfiglflda— i5) 4. Let X and Y be continuous random variables with joint density function misery where c is a constant. (a) (Spts) ind the value of c. "E l g t {‘Stos’lrxyawlamlt‘) =l 9? i[i(’§*u{3)mld3 L'P'l dim-v“ re) :l 5) if “3:, I”) C: % (b) (5pts) Find the marginal density fX _ “=76 “9 i 3 _ 5fxm~ 30(3-+«§—'g)c1\aw 3+7? o<g4g (C) (5pts) Find the conditional probability P(Y < < .L » » J. g :- o:- = .1 P(X< a) g” [Q n_ _ ‘riz-z, *Jfl_fl3~___Ju PI Xdin ‘10:?) ‘7 2? gait???“ “19’ 51 1" 3:). " 8‘ D ‘9- . J. ' .L . l , c 'C Mew: - P” .2 W. L (d) (5pts) Are X and Y independent? Justify your answer. X I E: . i , a, I 7 r We): SO(%+%§)M ~= 412+??? we; a Digué‘wstyd 5.(10pts) Let X1,X2,...,X25 be a random sample drawn from a normal population with mean ,u and standard deviation 0.04. What is the probability distribution of the sample mean X? Find the probabifity that the sample mean X will deviate from the true mean ,u by at most 0.02. {a xv (VC/UH—fl—Uojgl) ‘9 Pu 357*! s = W 557M 0:9.L < :P[”'§fi+$%x ) g = Peas; s 2 s 2.5) .g- 2 n firm) SI: gxmy 03) :7 it ‘1 are not Imnahivehcunij 6. The joint probability mass function two discrete random variables X and Y is given by 1 1 1 1 1,1:M 1,2:- 2,1:- , =—. f()8f()4f()8f(22)2 (a) (6pts) Compute the conditional mass function iny(9:|Y = 1). g . X; l ) txw'xlw): [>(x=vcl‘r:1) «r. WPiX;C:»E=') .: Pm) ._ fl Stow) ‘ "‘ "' %% _ _§+L :4wamu 3 {L n} «=4 2 #— u... (b) (Gpts) Compute P(X + Y > 2). pm— Y > .2 ) "‘ lime) + gun) + $2.2) 1m. ".J..._1. 4+J§f3'y 7.(12pts) The lifetime X of a certain brand of light bulb is a normally distributed random variable. 10% of all such light bulbs produced have a lifetime that exceeds 1.5 years. 5% of all such light bulbs produced have a lifetime that is less than 1 year. What are the mean and standard deviation of X '3’ W 0&1) r h :. (Ni ‘7 l" g- 0, 0‘ i '7 chlvétl):btf 3) 1%:4'69-5- “a? fit; MSH 6= 0.170? 8.(12pts) The sample mean X of a random sample of size n is used to estimate the true mean ,u of an infinite population that has standard deviation 0‘ = 0.1. What is the minimum sample size n required to ensure that the probability the sample mean X. is within 0.02 of the true mean to be at least 0.95, i.e.1 P(|X — m S 0.02) 2 0.95? l)[’0!9$ ‘5: 52")“ £033) will; '1) J; 3/ _ . 2. " 1: ...
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midterm2solution - AMS 310: Survey of Probability and...

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