hw6sol - 36-225 - Homework 6 SOLUTIONS + Grading Key1.5...

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Unformatted text preview: 36-225 - Homework 6 SOLUTIONS + Grading Key1.5 points: 3 for findingE(X2), 2 for V(X)LetXU(a,b). We already know thatE(X) =a+b2. ThenE(X2) =Zbax21b-adx=1b-ax33ba=1b-ab3-a33=1b-a(b-a)(b2+ab+a2)3=a2+ab+b23=V(X) =E(X2)-E2(X) =a2+ab+b23-(a+b)24=b2-2ab+a212=(b-a)2122.6 points: 2 for each part.(a)P(475< Y <500) =Z5004751500dy=1500y500475=500-475500=120(b)P(0< Y <25) =Z251500dy=25500=120(c)P(0< Y <250) =Z2501500dy=250500=123.4 points: 2 forE(A), 2 forV(A)RU(0,1) =fR(r) = 1 0< r <1Therefore,E(A) =E(R2) =Z1r21dr=r331=3(1.05).V(A) =V(R2) =E(R2)2-E2(R2) =E(2R4)-29.Now,E(2R4) =Z12r41dr=2r551=25.And so,V(A) =V(R2) =25-29=4245(.88).14.5 points: 3 for standardizing, 2 for using the tableLetYbe the amount of money spent on maintenance. ThenYN(400,20).P(Y >450) =P(Y->450-40020) =P(Z >2.5) =.00625.6 points: 3 for set-up, 3 for workNeed to findcsuch thatP(Y > c) = 0.1..1 =P(Y > c) =P(Y->c-40020) =P(Z >c-40020)The closest number in the body of the table to 0.1 is 0.1003 which is the table entry forz= 1.28. Then1.28 =c-40020=c= (1.28)(20) + 400 = 425.66.11 points: 4 for *, 6 for **, 1 for putting everything togetherXN(0,1)fX(x) =12ex22(- x )E(e3X+e-X22) =*z}|{E(e3X)+**z}|{E(e-X22)*E(e3X)=MX(t= 3)sinceXN(0,1)z}|{=e3.0+3212=e4.5* *E(e-X2/2) =Z-ex2/212e-x2/2dx=Z-12e-x2dx=Z-12e-x2212|{z}almostN(=0, 2=12)dx=12Z-1212e-x2212|{z}N(=0, 2=12)=121 =12Putting it all together,E(e3X+e-X22) =e4.5+1227.12 points: 3 points for each partLetXbe the demand for water in the early afternoon.Xexp(= 100)....
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This note was uploaded on 07/15/2011 for the course STAT 36225 taught by Professor Tom during the Summer '09 term at Carnegie Mellon.

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hw6sol - 36-225 - Homework 6 SOLUTIONS + Grading Key1.5...

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