CHAPTER 5. p(x,y) x

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1 CHAPTER 5 Sction 5.. a. P(X, Y p(,. b. P(X and Y p(, + p(, + p(, + p(,.4 c. At last on hos is in us at both islands. P(X and Y p(, + p(, + p(, + p(,.7 d. B summing row probabilitis, p (.6,.4,.5 for,,, and b summing column probabilitis, p (.4,.8,.8 for,,. P(X p ( + p (.5. P(,., but p ( p ( (.6(.4.84., so X and Y ar not indpndnt.. a. p(, b. P(X and Y p(, + p(, + p(, + p(,.56 (.8(.7 P(X P(Y c. P( X + Y P(X and Y p(,. d. P(X + Y p(, + p(, + p(,.5. a. p(,.5, th ntr in th st row and st column of th joint probabilit tabl. b. P( X X p(, + p(, + p(, + p(, c. A { (, : + } { (, : + } P(A p(, + p(, + p(4, + p(, + p(4, + p(4, + p(, + p(, + p(,. d. P( actl 4 p(, + p(, + p(, + p(4,.7 P(at last 4 P(actl 4 + p(4, + p(4, + p(4, + p(, + p(, + p(,.46 75

2 Chaptr 5: Joint Probabilit Distributions and Random Sampls 4. a. P ( P(X p(, + p(, + p(, + p(,.9 P ( P(X p(, + p(, + p(, + p(,., tc. 4 p ( b. P ( P(X p(, + p(, + p(, + p(, + p(4,.9, tc p ( c. p(4,, t p (4. > and p (.9 >, so p(, p ( p ( for vr (,, and th two variabls ar not indpndnt. 5. a. P(X, Y P( customrs, ach with packag P( ach has packag customrs P( customrs (.6 (.5.54 b. P(X 4, Y P(total of packags 4 customrs P(4 customrs Givn that thr ar 4 customrs, thr ar 4 diffrnt was to hav a total of packags:,,, or,,, or,,, or,,,. Each wa has probabilit (. (., so p(4, 4(. (.( a. p(4, P( Y X 4 P(X 4 (.6 (.4 ( b. P(X Y p(, + p(, + p(, + p(, + p(4,4.+(.(.6 + (.(.6 + (.5(.6 + (.5(

3 Chaptr 5: Joint Probabilit Distributions and Random Sampls c. p(, unlss,,, ;,,,, 4. For an such pair, p(, P(Y X P(X (. 6 (.4 p ( p (4 p( 4 p( 4, 4 p(4,4 (.6 4 ( p ( p(, + p(4, (.6 (.5 + (.6 (.4( p ( p(, + p(, + p(4, (.6 (. + (.6 (.4( (.6 (.4 ( p ( p(, + p(, + p(, + p(4, (. 6(. + (.6(.4(. 4 (.6(.4 (.5 + (.6(.4 (.5.59 p ( [ ] a. p(,. b. P(X and Y p(, + p(, + p(, + p(,. c. P(X p(, + p(, + p(,.; P(Y p(, + + p(5,. d. P(ovrflow P(X + Y > 5 P(X + Y 5 P[(X,Y(, or or (5, or (, or (, or (,] Th marginal probabilitis for X (row sums from th joint probabilit tabl ar p (.5, p (., p (.5, p (., p (4., p (5.; thos for Y (column sums ar p (.5, p (., p (.. It is now asil vrifid that for vr (,, p(, p ( p (, so X and Y ar indpndnt. 77

4 Chaptr 5: Joint Probabilit Distributions and Random Sampls , 6 a. numrator ( 56( 45(, 4,4 59,775 dnominator 775 ; p(,. 59 b. p(, ( +, _ ar _ non ngativ int grs _ such _ that + 6 othrwis 9. dd a. f (, dd K( + K K d dd + K dd K d + 9, K K 8, 6 6 ( d b. P(X < 6 and Y < 6 K + dd K c. 6 4K 6 8,4K.4 I + III II P( X Y rgion III f (, dd (, dd f f (, dd I 8 + II f (, dd (aftr much algbra.59 f (, dd 78

5 Chaptr 5: Joint Probabilit Distributions and Random Sampls d. f ( f (, d K ( + d K + K +.5, K. f ( is obtaind b substituting for in (d; clarl f(, f ( f (, so X and Y ar not indpndnt.. a. f(, 5 6,5 6 othrwis sinc f (, f ( for 5 6, 5 6 b. P(5.5 X 5.75, 5.5 Y 5.75 P(5.5 X 5.75 P(5.5 Y 5.75 (b indpndnc (.5(.5.5 c. 6 I +/6 / 6 5 II P((X,Y A A dd ara of A (ara of I + ara of II a. p(, µ µ!! for,,, ;,,, µ b. p(, + p(, + p(, [ + + µ ] c. P( X+Ym P( X k, Y m k m! ( + µ m m k m k µ k k k! ( m k m ( + µ m m k mk ( + µ k! µ k m! +. Poisson distribution with paramtr µ µ, so th total # of rrors X+Y also has a 79

6 Chaptr 5: Joint Probabilit Distributions and Random Sampls. (+ a. P(X> dd d. 5 b. Th marginal pdf of X is ( + d ( + (+ d th marginal pdf s, so th two r.v s ar not indpndnt. c. P( at last on cds P(X and Y d (+ dd for ; that of Y is for. It is now clar that f(, is not th product of ( d a. f(, f ( f (, othrwis b. P(X and Y P(X P(Y ( - ( -.4 ( c. P(X + Y dd [ ] d ( d. 594 ( d. P(X + Y [ ] d. 64 so P( X + Y P(X + Y P(X + Y , 4. a. P(X < t, X < t,, X < t P(X < t P( X < t ( t b. If succss {fail bfor t}, thn p P(succss t, and P(k succsss among trials ( k k t t k c. P(actl 5 fail P( 5 of l s fail and othr 5 don t + P(4 of l s fail, m fails, and othr t 5 ( + 4 t t 4 µ t t µ t 5 don t ( ( ( ( ( 8

7 Chaptr 5: Joint Probabilit Distributions and Random Sampls 8 5. a. F( P( Y P [(X ((X (X ] P (X + P[(X (X ] - P[(X (X (X ] ( ( ( + for b. f( F ( ( ( + ( ( 4 for E(Y ( 4 d 6. a. f(, (,, ( d k d f ( ( 7,, + b. P(X + X ( ( 7 d d (aftr much algbra.55 c. ( ( 7, ( ( d d f f a. (, ( Y X P within a circl of radius A R dd f A P, ( ( R A of ara dd πr A π b. π π, R R R Y R R X R P

8 Chaptr 5: Joint Probabilit Distributions and Random Sampls c. P R X R, R Y R R πr π f f (, d πr d d. ( R R R πr for R R and similarl for f Y (. X and Y ar not indpndnt sinc.g. f (.9R f Y (.9R >, t f(.9r,.9r sinc (.9R,.9R is outsid th circl of radius R. 8. a. P X ( rsults from dividing ach ntr in row of th joint probabilit tabl b p (.4: P P P.8 (.4. (.4.6 ( b. P X ( is rqustd; to obtain this divid ach ntr in th row b p (.5: P X (..8.6 c. P( Y P X ( + P X ( d. P X Y ( rsults from dividing ach ntr in th column b p (.8: P (

9 Chaptr 5: Joint Probabilit Distributions and Random Sampls 9. a. f f (, k( + ( f ( k +.5 Y X X k( + X ( k k +.5 8, f Y b. P( Y 5 X f Y 5 fy X 5 ( d k(( + d.78 5 k( +.5 P( Y 5 ( d (k +.5 d k(( + k( +.5 Y X ( c. E( Y X f d d 5.79 E( Y k(( + X d k( +.5 V(Y X E( Y X [E( Y X ] f (,,, f (, whr f, (, a. f (,, th marginal joint pdf f ( d,, of (X, X b. f (, f (,,, whr f ( f f (,, d ( d. For vr and, f Y X ( f (, sinc thn f(, f Y X ( f X ( f Y ( f X (, as rquird. 8

10 Chaptr 5: Joint Probabilit Distributions and Random Sampls Sction 5.. a. E( X + Y ( + p(, ( + (. + ( + 5( ( + 5(. 4. b. E[ma (X,Y] ma( + p(, ( (. + (5( (5( E(X X ( 4 p(, ( (.8 + ( ( (4 (.6.5 (which also quals E(X E(X Lt h(x,y # of individuals who handl th mssag. h(, Sinc p(, for ach possibl (,, E[h(X,Y] 84 h,. 8 ( 5. E(XY E(X E(Y L L L 6. Rvnu X + Y, so E (rvnu E (X + Y 5 ( + p(, p(, p(5,

11 Chaptr 5: Joint Probabilit Distributions and Random Sampls 7. E[h(X,Y] 6 dd ( 6 dd ( dd 5 6 d 8. E(XY p(, p( p ( p( p ( E(X E(Y. (rplac Σ with in th continuous cas µ µ. E(X 75 5 f ( d 4 ( d, so Var (X Similarl, Var(Y, so ρ X, Y Cov(X,Y and 5 5. a. E(X 5.55, E(Y 8.55, E(XY ((. + (( (5(. 44.5, so Cov(X,Y 44.5 (5.55( X Y, so ρ X, Y. 7 (.45(9.5 b..45, a. E(X f ( d [ K +.5] d 5.9 E( Y E(XY K( + dd Cov ( X, Y (5.9., so Var (X Var(Y ( b. E(X [ K +.5] d E( Y ρ..4 (8.664(

12 Chaptr 5: Joint Probabilit Distributions and Random Sampls. Thr is a difficult hr. Eistnc of r rquirs that both X and Y hav finit mans and variancs. Yt sinc th marginal pdf of Y is ( ( + for, E( d d ( + d ( d ( + + ( + first intgral is not finit. Thus r itslf is undfind., and th. Sinc E(XY E(X E(Y, Cov(X,Y E(XY E(X E(Y E(X E(Y - E(X E(Y, and sinc Corr(X,Y Cov( X, Y, thn Corr(X,Y 4. a. In th discrt cas, Var[h(X,Y] E{[h(X,Y E(h(X,Y] } [ h(, E( h( X, Y ] p(, with rplacing in th continuous cas. [ h(, p(, ] [ E( h( X, Y ] b. E[h(X,Y] E[ma(X,Y] 9.6, and E[h (X,Y] E[(ma(X,Y ] ( (. +(5 ( (5 (. 5.5, so Var[ma(X,Y] 5.5 ( a. Cov(aX + b, cy + d E[(aX + b(cy + d] E(aX + b E(cY + d E[acXY + adx + bcy + bd] (ae(x + b(ce(y + d ace(xy ace(xe(y accov(x,y b. Corr(aX + b, cy + d Cov( ax + b, cy + d accov( X, Y Var( ax + b Var( cy + d a c Var( X Var( Y Corr(X,Y whn a and c hav th sam signs. c. Whn a and c diffr in sign, Corr(aX + b, cy + d -Corr(X,Y. 6. Cov(X,Y Cov(X, ax+b E[X (ax+b] E(X E(aX+b a Var(X, so Corr(X,Y avar( X Var( X Var( Y avar( X Var( X a Var( X if a >, and if a < 86

13 Chaptr 5: Joint Probabilit Distributions and Random Sampls Sction P(..5. P( a p ( E ( (5(.4 +.5( ( µ b. s P(s.8... E(s.5 8. a. T 4 P(T b. µ E. µ T ( T c. T E( T E( T 5.8 (

14 Chaptr 5: Joint Probabilit Distributions and Random Sampls /n p(/n X is a binomial random variabl with p a. Possibl valus of M ar:, 5,. M whn all nvlops contain mon, hnc p(m (.5.5. M whn thr is a singl nvlop with $, hnc p(m p(no nvlops with $ ( p(m 5 [ ].87. M 5 p(m An altrnativ solution would b to list all 7 possibl combinations using a tr diagram and computing probabilitis dirctl from th tr. b. Th statistic of intrst is M, th maimum of,, or, so that M, 5, or. Th population distribution is a s follows: 5 p( / / /5 Writ a computr program to gnrat th digits 9 from a uniform distribution. Assign a valu of to th digits 4, a valu of 5 to digits 5 7, and a valu of to digits 8 and 9. Gnrat sampls of incrasing sizs, kping th numbr of rplications constant and comput M from ach sampl. As n, th sampl siz, incrass, p(m gos to zro, p(m gos to on. Furthrmor, p(m 5 gos to zro, but at a slowr rat than p(m. 88

15 Chaptr 5: Joint Probabilit Distributions and Random Sampls 4. Outcom,,,,4,,,,4 Probabilit r Outcom,,,,4 4, 4, 4, 4,4 Probabilit a r b. P( p ( c. r p(r d. P ( X.5 P(,,, + P(,,, + + P(,,, + P(,,, + + P(,,, + P(,,, + + P(,,, ( (.4 (. + 6(.4 (. + 4(.4 ( a p ( b p ( c. all thr valus ar th sam:.4 89

16 Chaptr 5: Joint Probabilit Distributions and Random Sampls 4. Th statistic of intrst is th fourth sprad, or th diffrnc btwn th mdians of th uppr and lowr halvs of th data. Th population distribution is uniform with A 8 and B. Us a computr to gnrat sampls of sizs n 5,,, and from a uniform distribution with A 8 and B. Kp th numbr of rplications th sam (sa 5, for ampl. For ach sampl, comput th uppr and lowr fourth, thn comput th diffrnc. Plot th sampling distributions on sparat histograms for n 5,,, and. 44. Us a computr to gnrat sampls of sizs n 5,,, and from a Wibull distribution with paramtrs as givn, kping th numbr of rplications th sam, as in problm 4 abov. For ach sampl, calculat th man. Blow is a histogram, and a normal probabilit plot for th sampling distribution of for n 5, both gnratd b Minitab. This sampling distribution appars to b normal, so sinc largr sampl sizs will produc distributions that ar closr to normal, th othrs will also appar normal. 45. Using Minitab to gnrat th ncssar sampling distribution, w can s that as n incrass, th distribution slowl movs toward normalit. Howvr, vn th sampling distribution for n 5 is not t approimatl normal. n Normal Probabilit Plot 9 Frqunc Probabilit n Andrson-D arling N ormalit Tst A-Squard: 7.46 P-Valu:. n 5 Normal Probabilit Plot Frqunc Probabilit Andrson- Darl ing Normali t Ts t A- Squa r d : P-Valu :. 9

17 Chaptr 5: Joint Probabilit Distributions and Random Sampls Sction µ cm.4 cm a. n 6 E( X cm.4 µ. cm n 4 b. n 64 E( X cm.4 µ. 5cm n 8 c. X is mor likl to b within. cm of th man ( cm with th scond, largr, sampl. This is du to th dcrasd variabilit of X with a largr sampl siz. 47. µ cm.4 cm.99. a. n 6 P(.99 X. P Z.. P(- Z Φ( - Φ( b. n 5 P( X >. Z >.4 / 5 P P( Z >.5 - Φ( µ X,. n P( X 5.5 P Z.. a. µ 5 P(-.5 Z b. P( X 5.5 P Z.. P(-.5 Z

18 Chaptr 5: Joint Probabilit Distributions and Random Sampls 49. a. P.M. 6:5 P.M. 5 minuts. With T X + + X 4 total grading tim, µ T nµ (4(6 4 and n, so P( T 5 P Z 5 4 P 7.95 ( Z T b. P ( T > 6 P Z > P( Z > µ, psi 5 psi a. n 4 9,9,,, P( 9,9 X, P Z 5/ 4 5/ 4 P(-.6 Z.5 Φ(.5 - Φ( b. According to th Rul of Thumb givn in Sction 5.4, n should b gratr than in ordr to appl th C.L.T., thus using th sam procdur for n 5 as was usd for n 4 would not b appropriat. 5. X ~ N(,4. For da, n 5 P( X P Z P( Z / 5 For da, n 6 P( X P Z P( Z / 6 For both das, P( X (.8686( X ~ N(, n 4 µ T nµ (4( 4 and n ((, T W dsir th 95 th prcntil: 4 + (.645( 4.9 9

19 Chaptr 5: Joint Probabilit Distributions and Random Sampls 5. µ 5,. a. n P( X 5 P Z P( Z / 9 b. n P( X 5 P Z P( Z 5.7. / µ X,. 7 n 5 a. µ P( X. P Z P( Z P(.65 X. P ( X. P( X b. P( X. P Z. 99 implis that.,.85/ n 85/ n which n.. Thus n will suffic. from µ npq np a. 4.5 P( 5 X P Z P(. Z b P( 5 X 5 P Z P (.59 Z a. With Y # of tickts, Y has approimatl a normal distribution with µ 5, , so P( 5 Y 7 Z Z P P( -.9 b. Hr µ 5, 5, Z , so P( 5 Y 75 P P( -.6 Z

20 Chaptr 5: Joint Probabilit Distributions and Random Sampls 57. E(X, Var(X, 4. 4 P( Z , so P( X 5 P Z 4.4 Sction a. E( 7X + 5X + 5X 7 E(X + 5 E(X + 5 E(X 7( + 5(5 + 5( 87,85 V(7X + 5X + 5X 7 V(X + 5 V(X + 5 V(X 7 ( + 5 ( + 5 (8 9,,6 b. Th pctd valu is still corrct, but th varianc is not bcaus th covariancs now also contribut to th varianc. 59. a. E( X + X + X 8, V(X + X + X 45, P(X + X + X P Z P( Z P(5 X + X + X P ( 4.47 Z µ X,. 6 n 55 6 P ( X 55 P Z P( Z P ( 58 X 6 P.89 Z.89. b. µ 6 ( 666 c. E( X -.5X -.5X ; V( X -.5X -.5X , sd P(- X -.5X -.5X 5 (..5 Z 5 P Z P

21 Chaptr 5: Joint Probabilit Distributions and Random Sampls d. E( X + X + X 5, V(X + X + X 6, P(X + X + X P Z P( Z W want P( X + X X, or writtn anothr wa, P( X + X - X E( X + X - X (6 -, V(X + X - X , 6, sd 8.8, so 4 ( P( X + X - X P Z P( Z Y is normall distributd with ( µ + µ ( µ + µ + µ µ Y, and 4 5 Y , Y ( Thus, P ( Y P Z P(.56 Z. 877 and.7795 P ( Y P Z P( Z a. Th marginal pmf s of X and Y ar givn in th solution to Ercis 7, from which E(X.8, E(Y.7, V(X.66, V(Y.6. Thus E(X+Y E(X + E(Y.5, V(X+Y V(X + V(Y.7, and th standard dviation of X + Y is.5 b. E(X+Y E(X + E(Y 5.4, V(X+Y 9V(X + V(Y 75.94, and th standard dviation of rvnu is E( X + X + X E( X + E(X + E(X min., V(X + X + X , Thus, P(X + X + X 6 P Z P( Z a. E(X.7, E(X.55, E(X X p(,. E(X X - E(X E(X , so Cov(X,X b. V(X + X V(X + V(X + Cov(X,X (

22 Chaptr 5: Joint Probabilit Distributions and Random Sampls 64. Lt X,, X 5 dnot morning tims and X 6,, X dnot vning tims. a. E(X + + X E(X + + E(X 5 E(X + 5 E(X 6 5(4 + 5(5 45 b. Var(X + + X Var(X + + Var(X 5 Var(X + 5Var(X c. E(X X 6 E(X - E(X Var(X X 6 Var(X + Var(X d. E[(X + + X 5 (X X ] 5(4 5(5-5 Var[(X + + X 5 (X X ] Var(X + + X 5 + Var(X X ] µ 5.,. E V ( X Y +.,. 566 X Y 5 5 a. ( X Y ; (. X Y. P(.77 Z P (b th CLT b. V ( X Y +.,. 47 X Y 6 6 P. X Y. P. Z.. ( ( a. With M 5X + X, E(M 5( + (4 5, Var(M 5 (.5 + ( 6.5, M b. P( 75 < M P < Z P(.4 < Z. 75 c. M A X + A X with th A I s and X I s all indpndnt, so E(M E(A X + E(A X E(A E(X + E(A E(X 5 d. Var(M E(M [E(M]. Rcall that for an r.v. Y, E(Y Var(Y + [E(Y]. Thus, E(M E ( A X + AX A X + A X E ( A E( X + E( A E( X E( A E( X + E( A E( X (b indpndnc (.5 + 5( (5(((4 + (.5 + ( , so Var(M (

23 Chaptr 5: Joint Probabilit Distributions and Random Sampls. E(M 5 still, but now Var ( M avar( X + a acov( X, X + avar( X (5(( Ltting X, X, and X dnot th lngths of th thr pics, th total lngth is X + X - X. This has a normal distribution with man valu + 5 4, varianc , and standard dviation.648. Standardizing givs P(4.5 X + X - X 5 P(.77 Z Lt X and X dnot th (constant spds of th two plans. a. Aftr two hours, th plans hav travld X km. and X km., rspctivl, so th scond will not hav caught th first if X + > X, i.. if X X < 5. X X has a man 5 5 -, varianc +, and standard dviation 4.4. Thus, 5 ( P ( X X < 5 P Z < P( Z < b. Aftr two hours, # will b + X km from whr # startd, whras # will b X from whr it startd. Thus th sparation distanc will b al most if X X, i.. X X, i.. X X. Th corrsponding probabilit is P( X X P(.4 Z a. E(X + X + X b. Assuming indpndnc of X, X, X, Var(X + X + X (6 + (5 + (8.5 c. E(X + X + X 4 as bfor, but now Var(X + X + X Var(X + Var(X + Var(X + Cov(X,X + Cov(X, X + Cov(X, X 745, with sd a. E (.5, so Y i n n n( n + E( W i E( Yi.5 i 4 i i n n n( n + (n + Var Y i so Var( W i Var( Yi.5 i 4 b. (.5, i i 97

24 Chaptr 5: Joint Probabilit Distributions and Random Sampls 7. W a. M a X + a X + W d a X + a X + 7, so E(M (5( + ((4 + (7(.5 58m M ( 5 (.5 + ( ( + ( 7 ( ,. 74 M 58 b. P ( M P Z P( Z Th total lapsd tim btwn laving and rturning is T o X + X + X + X 4, with E ( 4, 4, T o T o T o. T o is normall distributd, and th dsird valu t is th 99 th prcntil of th lapsd tim distribution addd to A.M.: : + [4+(5.477(.] : a. Both approimatl normal b th C.L.T. b. Th diffrnc of two r.v. s is just a spcial linar combination, and a linar combination X Y has approimatl a normal 8 6 µ and +.69,. 6 X Y X Y XY 4 5 of normal r.v s has a normal distribution, so distribution with P X Y & P Z.6.6 P (.7 Z c. ( 5 P & This probabilit is.6 µ d. ( X Y P Z P( Z.8.. quit small, so such an occurrnc is unlikl if µ 5, and w would thus doubt this claim. 74. X is approimatl normal with (5(.7 5 µ and (5(.7(.. 5 is Y with µ and. Thus 5 X Y p X Y 5 P Z X Y., so µ and 5 ( 5 P(. Z. 486, as 98

25 Chaptr 5: Joint Probabilit Distributions and Random Sampls Supplmntar Erciss 75. a. p X ( is obtaind b adding joint probabilitis across th row labld, rsulting in p X (.,.5,. for, 5, rspctivl. Similarl, from column sums p (.,.5,.55 for, 5, rspctivl. b. P(X 5 and Y 5 p(, + p(,5 + p(5, + p(5,5.5 c. p ( p ( (.(..5 p(,, so X and Y ar not indpndnt. (Almost an othr (, pair ilds th sam conclusion. d. E ( X + Y ( + p(,. 5 (or E(X + E(Y.5. E ( X Y + p(, Th roll-up procdur is not valid for th 75 th prcntil unlss 77. and or or both, as dscribd blow. Sum of prcntils: µ + Z + µ + ( Z µ + µ + ( Z( + Prcntil of sums: ( µ + + µ + ( Z Ths ar qual whn Z (i.. for th mdian or in th unusual cas whn + +, which happns whn. + or or both and + a. f (, dd kdd+ 8,5 k k 8,5 kdd b. f X ( kd k(5 kd k(45 + and b smmtr f Y ( is obtaind b substituting for in f X (. Sinc f X (5 >, and f Y (5 >, but f(5, 5, f X ( f Y ( f(, for all, so X and Y ar not indpndnt. 99

26 Chaptr 5: Joint Probabilit Distributions and Random Sampls 5 c. P ( X + Y 5 kdd+ 5 5 kdd, ,5 4 d. E X + Y E( X + E( Y { k( 5 + ( } k 45 d k ( d + (5, E ( XY f (, dd + k k dd k dd,5, 6.4 Cov(X,Y 6.4 ( , and E(X E(Y 4.654, so ( and ρ f. Var (X + Y Var(X + Var(Y + Cov(X,Y 7.66, so 78. F Y ( P( ma(x,, X n P( X,, X n [P(X ] n. n for. n n n E n n n n u du n n + n + n Thus f Y ( ( n ( d ( u + n ( Y u du n n n for 79. E ( X + Y + Z Var ( X + Y + Z + +.4, and th std dv P ( X + Y + Z 5 P( Z 9.

27 Chaptr 5: Joint Probabilit Distributions and Random Sampls 8. a. Lt X,, X dnot th wights for th businss-class passngrs and Y,, Y 5 dnot th tourist-class wights. Thn T total wight X + + X + Y + + Y 5 X + Y E(X E(X ( 6; V(X V(X (6 4. E(Y 5E(Y 5(4 ; V(Y 5V(Y 5( 5. Thus E(T E(X + E(Y And V(T V(X + V(Y , std dv b. P ( T 5 P Z P( Z a. E(N µ ((4 4 minuts b. W pct componnts to com in for rpair during a 4 hour priod, so E(N µ (( X ~ Bin (,.45 and Y ~ Bin (,.6. Bcaus both n s ar larg, both X and Y ar approimatl normal, so X + Y is approimatl normal with man ((.45 + ((.6 7, varianc (.45(.55 + (.6(.4.4, and standard dviation.. Thus, P(X Y 5 P Z P( Z P( µ. X µ +. & P Z./ n./ n P(. n Z. n, but (.96 Z n 97. n Th C.L.T. P so 84. I hav 9 oz. Th amount which I would consum if thr wr no limit is T o X + + X 4 whr ach X I is normall distributd with µ and. Thus T o is normal with µ 8 and 7. 48, so P(T o < 9 P(Z < T o T o 85. Th pctd valu and standard dviation of volum ar 87,85 and 47.7, rspctivl, so, 87,85 P ( volum, P Z P( Z Th studnt will not b lat if X + X X, i.. if X X + X. This linar combination has man, varianc 4.5, and standard dviation.6, so ( P ( X X + X P Z P( Z

28 Chaptr 5: Joint Probabilit Distributions and Random Sampls 87. a. Var( ax + Y a + acov( X, Y + a + a ρ + Substituting a Y ilds + Y ρ + Y Y ( ρ X Y, so ρ X Y. b. Sam argumnt as in a c. Suppos ρ. Thn Var ( ax Y Y ( ρ, which implis that ax Y k (a constant, so ax Y ax k, which is of th form ax + b. 88. X + Y t ( + t E ( f (, dd. To find th minimizing valu of t, tak th drivativ with rspct to t and quat it to : ( + t( f (, tf (, dd t ( + f (, dd E( X + Y, so th bst prdiction is th individual s pctd scor ( a. With Y X + X, F Y ( ν / ( ν / Γ( ν / ν / Γ But th innr intgral can b shown to b qual to [ ( ν / + ν Γ ( ν + ν / ν ν + ( ν+ ν / ] /, from which th rsult follows. d d. b. B a, ν Z + is chi-squard with Z, tc, until ν, so ( Z Z + Z + 9s chi-squard withν n Z... + Z n + is chi-squard with c. X i µ is standard normal, so is chi-squard with ν n. X i µ is chi-squard with ν, so th sum

29 Chaptr 5: Joint Probabilit Distributions and Random Sampls 9. a. Cov(X, Y + Z E[X(Y + Z] E(X E(Y + Z E(XY + E(XZ E(X E(Y E(X E(Z E(XY E(X E(Y + E(XZ E(X E(Z Cov(X,Y + Cov(X,Z. b. Cov(X + X, Y + Y Cov(X, Y + Cov(X,Y + Cov(X, Y + Cov(X,Y (appl a twic a. V X V ( W + E + V ( W + E V ( and ( W E X ( X, X Cov( W + E, W + E Cov( W, W + Cov( W, E ( E, W + Cov( E, E Cov( W, W V ( W w. Cov Cov ρ + Thus, W + E W W + E W + W E b. ρ a. Cov(X,Y Cov(A+D, B+E Cov(A,B + Cov(D,B + Cov(A,E + Cov(D,E Cov(A,B. Thus Corr( X, Y A Cov( A, B + + A D Cov( A, B A B A B + + D B B E E Th first factor in this prssion is Corr(A,B, and (b th rsult of rcis 7a th scond and third factors ar th squar roots of Corr(X, X and Corr(Y, Y, rspctivl. Clarl, masurmnt rror rducs th corrlation, sinc both squar-root factors ar btwn and. b rducs th corrlation.. This is disturbing, bcaus masurmnt rror substantiall

30 Chaptr 5: Joint Probabilit Distributions and Random Sampls 9. E Y & h( µ, µ, µ, µ [ + + ] 6 ( 4 5 Th partial drivativs of µ, µ, µ, 4 4,, and 4 h ( µ 4 with rspct to,,, and 4 ar, + +, rspctivl. Substituting, 5,, and 4 givs., -.5, -., and.67, rspctivl, so V(Y ((-. + ((-.5 + (.5(-. + (4.( , and th approimat sd of is Th four scond ordr partials ar, E(Y , 4, and rspctivl. Substitution givs 4