347.2010.final.sol

# 347.2010.final.sol - NAME 55L U 7Z/ﬂ/K ACSC/MATH 347/447...

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Unformatted text preview: NAME 55L U 7Z/ﬂ/K/ ACSC/MATH 347/447 Final Exam May 11, 2010 Directions. Show all work. Full credit will not be given unless suﬁicient work is shown or a suitable explanation given. You may not use the cdf or pdf functions on a calculator in any problem. You may leave binomial coefﬁcients in your answers. The maximum score is 100 points, and 21/2 hours is allowed for the test. 1. (12 points) HA and B are events With P(A) = 0.2, P(B) = 0.3, and P(A n B) = 0.1, ﬁnd: a. 10(2) :: 1’P(A) = 1.40.; =0. 5’ b. P(AUB) = P(A)+P(BI'F(AAB) = 0.1+a.3- 04 = o. If c.P(A|B) = Fiﬁ/"Bl _ 9_-_L .. _L PCB) " o~3 ' 3 d. PJmB) : P(B)’P(ﬂ/73J = 0.3—0.|=a.1 e. P(ZU§) = P( m) "-’— fl 'PIADB/ = 1-O._1:O.9 f. Are A and B independent? You must explain for credit. no, sc‘hce I’M—HZ)? PM) «1 w‘m 64:9(AAB)#P(A)P(B’=(0-1)(0-3) ACSC/MATH 347/447 Final Exam May 11, 2010 Page 2 of 9 2. (5 points) A certain kind of appliance requires repairs on the average once every 3 years. Assuming that the times between repairs are exponentially distributed, what is the probability such an appliance will work at least 5 years without requiring repairs? 3. (5 points) In the inspection of a fabric produced in continuous rolls, there is, on average, one imperfection per yard. Assuming that the number of imperfections is a random variable having a Poisson distribution, ﬁnd the probability that 5 yards of the fabric will have at most 2 imperfections. La ‘/ '—’ #eg’WwS‘Wtf ﬁW. MYNKMVK(\$°1=§)WM ”Ye-.1): arm-rout) +6”) - ,1 ." '9' = 5°c 5+ .s e 5 ‘ 0! 1'. r-r 2.! -— 8'51 1+s’+ 25') ACSC/MATH 347/447 Final Exam May 11, 2010 Page 3 of9 4. (10 points) A random variable, Y, has a normal distribution with mean 5 and standard deviation 10. a. Find P(4 S Y3 7). P(es‘rs7)=P(”,':—' s £5 7:5) .-: p(-—o.| a—Zg 0.2) ﬂ? 7w'ffabfe‘. _______.__..__ =1—P(2‘>0.2)-P(a>o.u) = 1'0.’+107 -(),L[éol —.= OJICH n M— soA Tulane". = Hz<a.;}-p(24-a.,3 = P(Z < 0-23 — [1 —P(zsa..)] =o.5773 -L1-o.;373>1= 0.”?! b. Find the value c so that P(Y S c) = 0.025. Be careful; where does c lie relative to the mean of Y? Text TaNL.‘ P(vec)=o.o 2.; => PM 2 — c) = 0. 013‘ .. -c+S’ .. - .) P(Z} ’6 —0.015 ACSC/MATH 347/447 Final Exam May 11, 2010 Page 4 of 9 5. (9 points) Let Y1 and Y2 have: means 2 and 3, respectively; variances 5 and 9, respectively; and COV(Y1, Y2) 2—2. a. Find the mean and the variance of the random variable U 1 = — 3Y1 + 5. E(U‘):E(-3Y'+\$)='SE(V()+\$:'3(1J +5.1; V014) '—’ \/(’3Y:“’.5’) '3 V(’3Y.>=(-3)iV/‘/,) '-'—’ ‘? (9):; b. Find the mean of the random variable U 2 = Y22. Ewan = 5015): Wm ﬂab/gr =9+31 = [5’ c. Find the mean and the variance of the random variable U 3 = 7Y1 — 4Y2. Elu,)=5(7‘/, “(YA = 750/: F7503) = 7(1)‘ *(3): 9x. P V(U3)-:\/(7Y, " ‘fYaJ " 7‘ VM +(-‘I)'”V(Yz) *Mm'l’xwm 2.)? ___= 559 (5')-+ (6(9 ) + 1(73("")('1) ' : 1. 5'0 ACSC/MATH 347/447 Final Exam May 11, 2010 Page 5 of9 6. (10 points) Let the distribution function (cdf) of a continuous random variable Y be 0, ySO 2 y?’ 0<y32 a. Find P(1£YS3). Fm: 8y_y2_8 b. Find P(Y§3|Y21). 2 S4 8 , <y c. Find the density function (pdf) of Y . 1 y24. 9—1 Nieves) =r-=(3)—r=u) -8(3)'3"—3’ I1,2-J—=.§ ' 8 “3"5"? 6’ ' ' " " 9(V21) “ 7/? .=-__Z—_ 2. sum my: 1)=1-P(‘/-‘—1)=1‘1§' =35 (C3 3. i ....—— i- - ) \$(y)=F'((yj: 9") " My 74": 1- £ ) l‘JﬁL/ ACSC/MATH 347/447 Final Exam May 11, 2010 Page 6 of9 7. (12 points) Let Y1 and Y2 be continuous random variables with joint density function %(2y1+y2), 0<y1<1, 0<y2 <2 . Draw (and use) a picture ! 0, elsewhere f(y1,y2)={ a. Find P(Y2§ Y1). b. Find the marginal density function, f2 0/2), of Y2. ‘1 __ ‘6'} c. Find the conditional density function, f ( 321 | l ), of Y1, given Y2 = l. ‘6 I l (39 I’D/19(5): Sftlyumam 0 a i A 1 .. ’- _ 5- 1. '01 li’VWﬂﬂggift‘lﬂ'Jc-i §’i/I 1 = £13 1 __ s- ;th a - 55, m 2 (5:) 5’1(7*)=i&§'(711%)491 l 0<7}<3_ 1 V; :L ACSC/MATH 347/447 Final Exam May 11, 2010 Page 7 of 9 X, 2 s y s 4 8. (16 points) Let Y be a random variable with density function f (y) = 6 0, a. Find the distribution function, F (y), of Y. elseWhere' b. Find the mean of Y. c. Find the support of U = 3Y— 2; that is, ﬁnd the set of all real numbers u for which the density function of U is positive. (1. Find the density function of U = 3 Y — 2 using either the distribution function method or the method of transformations. e. (4 points extra credit) Find the density function of U both ways. _ _ °<> 3L z £4 ’1 9. 60 0 <; F[ )3 1. ) 0V 7 ﬁi) Léiélf i) y >7 H - an. L} 3 ’7‘ _ Ci? L:(Y):;eiﬁy)ﬂ% :£#[%)0% 4;; = 673“”? :v SWH/Uecgui3°2~léM93“f’l§~={MJLfé(/< 2m} ACSC/MATH 347/447 Final Exam May 11, 2010 Page 8 of9 9. (12 points) I have a drawer that contains 10 white, 6 blue, and 4 brown socks. a. If I select 2 socks at random without replacement, ﬁnd the probability that I get two of the same color. (C2)-f(&)—f Haunaglzmed: (”I ‘fS"-rls'-r 6 _ 33 _. _____._.__—-——— .. ”—0.397 I70 95 b. If I select 6 socks at random with replacement, ﬁnd the probability I get exactly two brown socks. (ﬁost-GWSMS Mm 6 W, Yrs/64,... (6’10“.3") 1 0' PW =1) —_ (905371 g)“ —-= a, 14:7; C IN select socks at random with replacement ﬁnd the probability it takes me at least 3 tries to get a brown sock. d. (4 points extra credit) If I select 6 socks at random with replacement, ﬁnd the probability that I get 3 white, 1 blue, and 2 brown socks in some order. (567,: :14wa Yﬁ #06,... 73'54W “W (Vt)\/w\/3) NM?“ WJ(6)JO’£- )0 3?; :40 - 6‘. 1 Mann-3,1,1, 3-) (£367) -.— 40(4) :36}- ~—)(3—)" = 60(0.5')3(o.3)(a.2) z 0.0; ACSC/MATH 347/447 Final Exam May 11, 2010 Page 9 of 9 10. (9 points) Let Y1 and Y2 be discrete random variables with joint probability function 19021, y2) given by the table below. a. Find the probability function of the marginal distribution of Y1 . b. Find E(Y1)and V(Y1). c. (6 points extra credit) Find the covariance of Y1 and Y2. MIIII-_ Ilﬂﬂ-ﬂ-l- —IIII-— —IIII- 0 so I 91) C99. ODL(&’L> @W Lme%f7\a W My» M3, 5“ W (4/ E(‘/u:3;awa>/7ti:gia>r MW ” l 2 y 1 : 7. ENE.) 33%; £3 0M%) OFWW W viii) : P(\/Z)_[E(\/¢)jz: (73057—2352: 0'? 75 _&_ . £9 Emu) 3333‘? 3 WWW) :, o +1~(9J0c‘f0+0140+04;g wig-1.0 as+3o0Js+Ml~§ 2 30'7 W01 v3) - ECYa Y2) E63) 25(71):;2 7 (2 25 )(l)=ﬁ_ 40.15 ...
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