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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‘ZdLGriilg> S41CrxtiM;,ﬂ,7 og/gﬁa ? 6:}?
gift: = Em; 1W” ~33: 4311;: 5%
UMCX): “$3 “ (“é“2.: 2. The time X (in hours) required to repair a machine is an exponential r.v. with parameter A = %. Find
(at) (613135) the probability that a repair time exceeds 2 hours? mil'l' I
2; I P(X>l)= e = e (b) (6pts) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds
9 hours? MLa 7iOIX7‘?): P(X>f) : e“; 3.(10pts) A randomly chosen IQ test taker obtains a score that is normally distributed with mean 100 and
standard deviation 15. What is the probability that the score of such a person is between 90 and 110. pwosxeno : H‘lﬁglﬂda— i5) 4. Let X and Y be continuous random variables with joint density function misery where c is a constant.
(a) (Spts) ind the value of c. "E l g t
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(b) (5pts) Find the marginal density fX _ “=76 “9 i 3 _
5fxm~ 30(3+«§—'g)c1\aw 3+7? o<g4g
(C) (5pts) Find the conditional probability P(Y < < .L
» » J. g : o: = .1
P(X< a) g” [Q
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(d) (5pts) Are X and Y independent? Justify your answer. X I E:
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We): SO(%+%§)M ~= 412+??? we; a Digué‘wstyd 5.(10pts) Let X1,X2,...,X25 be a random sample drawn from a normal population with mean ,u and
standard deviation 0.04. What is the probability distribution of the sample mean X? Find the probabiﬁty that the sample mean X will deviate from the true mean ,u by at most 0.02. {a xv (VC/UH—ﬂ—Uojgl) ‘9 Pu 357*! s = W 557M 0:9.L < :P[”'§ﬁ+$%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)
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3 {L n} «=4 2 #—
u... (b) (Gpts) Compute P(X + Y > 2).
pm— Y > .2 ) "‘ lime) + gun) + $2.2) 1m. ".J..._1.
4+J§f3'y 7.(12pts) The lifetime X of a certain brand of light bulb is a normally distributed random variable. 10%
of all such light bulbs produced have a lifetime that exceeds 1.5 years. 5% of all such light bulbs produced
have a lifetime that is less than 1 year. What are the mean and standard deviation of X '3’ W 0&1) r h :. (Ni ‘7 l" g 0, 0‘ i
'7 chlvétl):btf 3) 1%:4'695
“a? ﬁt; MSH 6= 0.170? 8.(12pts) The sample mean X of a random sample of size n is used to estimate the true mean ,u of an
inﬁnite population that has standard deviation 0‘ = 0.1. What is the minimum sample size n required to ensure that the probability the sample mean X. is within 0.02 of the true mean to be at least 0.95, i.e.1
P(X — m S 0.02) 2 0.95? l)[’0!9$ ‘5: 52")“ £033) will; '1) J; 3/ _ . 2. "
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 Spring '08
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