PART ONE. Solutions to Exercises

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Transcription:

PART ONE Soutos to Exercses

Chapter Revew of Probabty Soutos to Exercses 1. (a) Probabty dstrbuto fucto for Outcome (umber of heads) 0 1 probabty 0.5 0.50 0.5 Cumuatve probabty dstrbuto fucto for Outcome (umber of heads) 0 0 1 1 Probabty 0 0.5 0.75 1.0 (c) = E ( ) (0 0.5) (1 0.50) ( 0.5) 1.00 Usg Key Cocept.3: var( ) E( ) [ E( )], ad E ( ) (0 0.5) (1 0.50) ( 0.5) 1.50 so that var( ) E( ) [ E( )] 1.50 (1.00) 0.50.. We kow from Tabe. that Pr( 0) 0, Pr( 1) 0 78, Pr( 0) 0 30, Pr( 1) 0 70. So (a) E( ) 0 Pr ( 0) 1 Pr ( 1) 0 0 1 0 78 0 78, E( ) 0 Pr ( 0) 1 Pr ( 1) 0 0 30 1 0 70 0 70 E[( ) ] (0 0.70) Pr ( 0) (1 0.70) Pr ( 1) ( 0 70) 0 30 0 30 0 70 0 1 E[( ) ] (0 0.78) Pr ( 0) (1 0.78) Pr ( 1) ( 0 78) 0 0 0 78 0 1716 (c) Tabe. shows Pr( 0, 0) 0 15, Pr( 0, 1) 0 15, Pr( 1, 0) 0 07, Pr( 1, 1) 0 63. So

4 Stock/Watso - Itroducto to Ecoometrcs - Bref Edto cov(, ) E[( )( )] (0-0.70)(0-0.78)Pr( 0, 0) (0 0 70)(1 0 78)Pr ( 0 1) (1 0 70)(0 0 78)Pr ( 1 0) (1 0 70)(1 0 78)Pr ( 1 1) ( 0 70) ( 0 78) 0 15 ( 0 70) 0 0 15 0 30 ( 0 78) 0 07 0 30 0 0 63 0 084, 0 084 cor (, ) 0 445 0 1 0 1716 3. For the two ew radom varabes W 3 6 ad V 0 7, we have: (a) (c) 4. (a) WV E( V) E(0 7 ) 0 7 E( ) 0 7 0 78 14 54, E( W) E(3 6 ) 3 6 E( ) 3 6 0 70 7 var(3 6 ) 6 36 0 1 7 56, W var(0 7 ) ( 7) 49 0 1716 8 4084 V cov (3 6, 0 7 ) 6( 7) cov(, ) 4 0 084 3 58 WV 3 58 cor ( W, V ) 0 445 7 56 8 4084 3 3 3 E( ) 0 (1 p) 1 p p k k k E( ) 0 (1 p) 1 p p (c) E ( ) 0.3 W var( ) E( ) [ E( )] 0.3 0.09 0.1 Thus, 0.1 0.46. V To compute the skewess, use the formua from exercse.1: Ateratvey, Thus, skewess E( ) E( ) 3[ E( )][ E( )] [ E( )] 3 3 3 3 0.3 3 0.3 0.3 0.084 3 3 3 E ( ) = [(1 0.3) 0.3] [(0 0.3) 0.7] 0.084 3 3 3 E ( ) /.084/0.46 0.87.

Soutos to Exercses Chapter 5 To compute the kurtoss, use the formua from exercse.1: Ateratvey, Thus, kurtoss s E( ) E( ) 4[ E( )][ E( )] 6[ E( )] [ E( )] 3[ E( )] 4 4 3 4 3 4 0.3 4 0.3 6 0.3 3 0.3 0.0777 4 4 4 E ( ) = [(1 0.3) 0.3] [(0 0.3) 0.7] 0.0777 4 4 4 E ( ) / =.0777/0.46 1.76 5. Let deote temperature F ad deote temperature C. Reca that 0 whe 3 ad 100 whe 1; ths mpes (100/180) ( 3) or 17.78 (5/9). Usg Key Cocept.3, 70 F mpes that 17.78 (5/9) 70 1.11 C, ad 7 F mpes (5/9) 7 3.89 C. 6. The tabe shows that Pr( 0, 0) 0 045, Pr( 0, 1) 0 709, Pr( 1, 0) 0 005, Pr( 1, 1) 0 41, Pr( 0) 0 754, Pr( 1) 0 46, Pr( 0) 0 050, Pr( 1) 0 950. (a) Uempoymet Rate (c) Cacuate the codtoa probabtes frst: The codtoa expectatos are E( ) 0 Pr( 0) 1 Pr( 1) 0 0 050 1 0 950 0 950 #(uempoyed) #(abor force) Pr( 0) 0 050 1 0 950 1 E( ) Pr( 0, 0) 0 045 Pr( 0 0) 0 0597, Pr( 0) 0 754 Pr( 0, 1) 0 709 Pr( 1 0) 0 9403, Pr( 0) 0 754 Pr( 1, 0) 0 005 Pr( 0 1) 0 003, Pr( 1) 0 46 Pr( 1, 1) 0 41 Pr( 1 1) 0 9797 Pr( 1) 0 46 E( 1) 0 Pr ( 0 1) 1 Pr ( 1 1) 0 0 003 1 0 9797 0 9797, E( 0) 0 Pr ( 0 0) 1 Pr ( 1 0) 0 0 0597 1 0 9403 0 9403 (d) Use the souto to part,

6 Stock/Watso - Itroducto to Ecoometrcs - Bref Edto Uempoymet rate for coege grads 1 E( 1) 1 0.9797 0.003. Uempoymet rate for o-coege grads 1 E( 0) 1 0.9403 0.0597. (e) The probabty that a radomy seected worker who s reported beg uempoyed s a coege graduate s Pr( 1, 0) 0 005 Pr( 1 0) 0 1 Pr( 0) 0 050 The probabty that ths worker s a o-coege graduate s Pr( 0 0) 1 Pr( 1 0) 1 0 1 0 9 (f) Educatoa achevemet ad empoymet status are ot depedet because they do ot satsfy that, for a vaues of x ad y, Pr( y x) Pr( y ) For exampe, Pr( 0 0) 0 0597 Pr( 0) 0 050 7. Usg obvous otato, C M F ; thus ad C M F C M F cov( MF, ). Ths mpes (a) C 40 45 $85,000 per year. (c) cor M F so that Cov ( M, F) cor ( M, F ). Thus Cov( M, F) (, ), M F M F Cov ( M, F ) 1 18 0.80 17.80, where the uts are squared thousads of doars per year. C M F cov( MF, ), so that C 1 18 17.80 813.60, ad C 813.60 8.54 thousad doars per year. (d) Frst you eed to ook up the curret Euro/doar exchage rate the Wa Street Joura, the Federa Reserve web page, or other faca data outet. Suppose that ths exchage rate s e (say e 0.80 euros per doar); each 1$ s therefore wth ee. The mea s therefore e C ( uts of thousads of euros per year), ad the stadard devato s e C ( uts of thousads of euros per year). The correato s ut-free, ad s uchaged. 8. E ( ) 1, 1 var( ) 4. Wth Z ( 1), m s Z 1 1 1 E ( 1) ( m 1) (1 1) 0, 1 1 1 var ( 1) s 4 1 4 4 Z

Soutos to Exercses Chapter 7 9. Vaue of Probabty 14 30 40 65 Dstrbuto of Vaue of 1 0.0 0.05 0.10 0.03 0.01 0.1 5 0.17 0.15 0.05 0.0 0.01 0.40 Probabty dstrbuto of 8 0.0 0.03 0.15 0.10 0.09 0.39 0.1 0.3 0.30 0.15 0.11 1.00 (a) The probabty dstrbuto s gve the tabe above. E ( ) 14 0.1 0.3 30 0.30 40 0.15 65 0.11 30.15 E ( ) 14 0.1 0.3 30 0.30 40 0.15 65 0.11 117.3 Var() E( ) [ E( )] 18.1 14.77 Codtoa Probabty of 8 s gve the tabe beow Vaue of 14 30 40 65 0.0/0.39 0.03/0.39 0.15/0.39 0.10/0.39 0.09/0.39 E( 8) 14 (0.0/0.39) (0.03/0.39) 30 (0.15/0.39) E 40 (0.10/0.39) 65 (0.09/0.39) 39.1 ( 8) 14 (0.0/0.39) (0.03/0.39) 30 (0.15/0.39) 40 (0.10/0.39) 65 (0.09/0.39) 1778.7 Var( ) 1778.7 39.1 41.65 8 15.54 (c) E( ) (1 14 0.0) (1 : 0.05) (8 65 0.09) 171.7 Cov(, ) E( ) E( ) E( ) 171.7 5.33 30.15 11.0 Corr(, ) Cov(, )/( ) 11.0/(5.46 14.77) 0.136 10. Usg the fact that f N m, s the ~ N (0,1) ad Appedx Tabe 1, we have (a) 1 3 1 Pr( 3) Pr F (1) 0 8413 Pr( 0) 1 Pr( 0) 3 0 3 1 Pr 1 F( 1) F(1) 0 8413 3 3

8 Stock/Watso - Itroducto to Ecoometrcs - Bref Edto (c) (d) 40 50 50 5 50 Pr(40 5) Pr 5 5 5 F (0 4) F ( ) F (0 4) [1 F ()] 0 6554 1 0 977 0 636 11. (a) 0.90 0.05 (c) 0.05 (d) Whe (e) ~ 10, the 10,, 6 5 5 8 5 Pr(6 8) Pr F ( 113) F (0 7071) /10 ~ F. 0 9831 0 760 0 9 Z where Z ~ N(0,1), thus Pr( 1) Pr( 1 Z 1) 0.3. 1. (a) 0.05 0.950 (c) 0.953 (d) The t df dstrbuto ad N(0, 1) are approxmatey the same whe df s arge. (e) 0.10 (f) 0.01 13. (a) E E W W ( ) Var ( ) 1 0 1; ( ) Var ( ) W 100 0 100. ad W are symmetrc aroud 0, thus skewess s equa to 0; because ther mea s zero, ths meas that the thrd momet s zero. 4 E ( ) (c) The kurtoss of the orma s 3, so 3 ; sovg yeds yeds the resuts for W. (d) Frst, codto o 0, so that S W : E( S 0) 0; E( S 0) 100, E( S 0) 0, E( S 0) 3 100 Smary, $ 3 4. 3 4 E( S 1) 0; E( S 1) 1, E( S 1) 0, E( S 1) 3. From the arge of terated expectatos E( S) E( S 0) Pr( 0) E( S 1) Pr( 1) 0 4 E( ) 3; a smar cacuato E( S ) E( S 0) Pr( 0) E( S 1) Pr( 1) 100 0.01 1 0.99 1.99 3 3 3 E( S ) E( S 0) Pr( 0) E( S 1) Pr( 1) 0 4 4 4 E( S ) E( S 0) Pr( 0) E( S 1) Pr( 1) 3 100 0.01 3 1 0.99 30.97

Soutos to Exercses Chapter 9 (e) ( ) 0, S ES thus Smary, E S E S from part d. Thus skewess 0. 3 3 ( S ) ( ) 0 E( S ) E( S ) 1.99, ad E S E S Thus, S S kurtoss 30.97/(1.99 ) 76.5 4 4 ( S ) ( ) 30.97. 14. The cetra mt theorem suggests that whe the sampe sze () s arge, the dstrbuto of the sampe average ( ) s approxmatey N, wth. Gve 100, 43 0, (a) 100, 43 0 43, ad 100 100 101 100 Pr( 101) Pr F (1 55) 0 9364 0 43 0 43 165, 43 0 606, ad 165 100 98 100 Pr( 98) 1 Pr( 98) 1 Pr 0 606 0 606 1 F ( 3 9178) F (3 9178) 1 000 (rouded to four decma paces) (c) 64, 43 0 6719, ad 64 64 101 100 100 103 100 Pr(101 103) Pr 0 6719 0 6719 0 6719 F (3 6599) F (1 00) 0 9999 0 8888 0 1111 15. (a) Pr(9.6 10.4) Pr Pr where Z ~ N(0, 1). Thus, 9.6 10 10 10.4 10 4/ 4/ 4/ 9.6 10 10.4 10 Z 4/ 4/ () 0; 9.6 10 10.4 10 Pr Z Pr( 0.89 Z 0.89) 0.63 4/ 4/ () 100; 9.6 10 10.4 10 Pr Z Pr(.00 Z.00) 0.954 4/ 4/ () 1000; 9.6 10 10.4 10 Pr Z Pr( 6.3 Z 6.3) 1.000 4/ 4/

10 Stock/Watso - Itroducto to Ecoometrcs - Bref Edto As get arge Pr(10 c 10 c) Pr c 4/ c 10 c 4/ 4/ 4/ c c Pr Z. 4/ 4/ gets arge, ad the probabty coverges to 1. (c) Ths foows from ad the defto of covergece probabty gve Key Cocept.6. 16. There are severa ways to do ths. Here s oe way. Geerate draws of, 1,,. Let 1 f 3.6, otherwse set 0. Notce that s a Berou radom varabes wth Pr( 1) Pr( 3.6). Compute. Because coverges probabty to Pr( 1) Pr( 3.6), w be a accurate approxmato f s arge. 17. 0.4 ad 0.4 0.6 0.4 (a) () P( 0.43) () P( 0.37) 0.4 0.43 0.4 0.4 Pr Pr 0.614 0.7 0.4/ 0.4/ 0.4/ 0.4 0.37 0.4 0.4 Pr Pr 1. 0.11 0.4/ 0.4/ 0.4/ 0.41 0.4 We kow Pr( 1.96 Z 1.96) 0.95, thus we wat to satsfy 0.41 1.96 ad 0.39 0.4 0.4/ 1.96. Sovg these equates yeds 90. 18. Pr( $ 0) 0 95, Pr( $ 0000) 0 05. (a) The mea of s The varace of s 0 Pr( $ 0) 0,000 Pr( $ 0000) $ 1000. 0.4/ E 19. (a) (0 1000) so the stadard devato of s Pr 0 (0000 1000) Pr( 0000) 7 ( 1000) 0 95 19000 0 05 1 9 10, 1 7 $ 7 (1 9 10 ) 4359 1 9 10 5 () E( ) m $ 1000, 1 9 10. 100 () Usg the cetra mt theorem, Pr( 000) 1 Pr( 000) 1 Pr 1000 000 1 000 1 9 10 1 9 10 5 5 1 F ( 94) 1 0 9891 0 0109

Soutos to Exercses Chapter 11 Pr ( y ) Pr ( x, y ) j j 1 1 Pr ( y x )Pr ( x ) j E ( ) y Pr( y ) y Pr( y x )Pr( x ) j j j j j 1 j 1 1 1 1 k k k j 1 y Pr( y x ) Pr( x ) j j E( x )Pr( x ) (c) Whe ad are depedet, so Pr ( x, yj ) Pr ( x)pr ( y j ) s E[( m )( m )] k 1 j 1 k 1 j 1 ( x m )( y m )Pr( x, y ) j j ( x m )( y m )Pr( x )Pr( y ) j j ( x m )Pr( x ) ( y m )Pr( y j j 1 j 1 E( m ) E( m ) 0 0 0, 0 cor (, ) 0 k 0. (a) m Pr( y ) Pr( y x, Z z )Pr( x, Z z ) j h j h j 1 h 1 k E( ) y Pr ( y )Pr ( y ) 1 y Pr ( y x, Z z )Pr ( x, Z z ) j h j h 1 j 1 h 1 m k j 1 h 1 1 j 1 h 1 k m m y Pr ( y x, Z z ) Pr ( x, Z z ) j h j h E( x, Z z )Pr ( x, Z z ) j h j h where the frst e the defto of the mea, the secod uses (a), the thrd s a rearragemet, ad the fa e uses the defto of the codtoa expectato.

1 Stock/Watso - Itroducto to Ecoometrcs - Bref Edto 1. (a) E E E 3 3 3 ( ) [( ) ( )] [ ] E( ) 3 E( ) 3 E( ) E( ) 3 E( ) E( ) 3 E( )[ E( )] [ E( )] 3 3 3 3 E( ) 3 E( ) E( ) E( ) 3 3 E E 4 3 3 ( ) [( 3 3 )( )] E 4 3 3 3 3 4 [ 3 3 3 3 ] E( ) 4 E( ) E( ) 6 E( ) E( ) 4 E( ) E( ) E( ) 4 3 3 4 E( ) 4[ E( )][ E( )] 6[ E( )] [ E( )] 3[ E( )]. The mea ad varace of R are gve by w 4 3 4 0.08 (1 w) 0.05 w 0.07 (1 w) 0.04 w (1 w) [0.07 0.04 0.5] where 0.07 0.04 0.5 Cov ( Rs, R b) foows from the defto of the correato betwee R s ad R b. (a) 0.065; 0.044 0.075; 0.056 (c) w 1 maxmzes ; 0.07 for ths vaue of w. (d) The dervatve of d dw wth respect to w s w.07 (1 w) 0.04 ( 4 w) [0.07 0.04 0.5] 0.010w 0.0018 sovg for w yeds w 18 /10 0.18. (Notce that the secod dervatve s postve, so that ths s the goba mmum.) Wth w 0.18, R.038. 3. ad Z are two depedety dstrbuted stadard orma radom varabes, so Z 0, Z 1, Z 0. (a) Because of the depedece betwee ad Z, Pr( Z z x) Pr( Z z ), ad (c) E( Z ) E( Z ) 0. Thus E ad E( ) E( Z ) E( ) E( Z ) 0 ( ) 1, E( Z) E( ) Z 1 0 1 3 3 ( ) ( ) ( ) ( ). 3 E E Z E E Z Usg the fact that the odd momets of a stadard orma radom varabe are a zero, we have E ( ) 0. Usg the depedece betwee ad Z, we 3 have E( Z ) 0. Thus E( ) E( ) E( Z ) 0. Z

Soutos to Exercses Chapter 13 (d) Cov( ) E[( )( )] E[( 0)( 1)] E( ) E( ) E( ) 0 0 0 0 cor (, ) 0 4. (a) E ( ) ad the resut foows drecty. ( / ) s dstrbuted..d. N(0,1), W ( / ), 1 ad the resut foows from the defto of a radom varabe. (c) E(W) (d) Wrte E( W) E E. 1 1 V / s 1 1 ( / s ) 1 1 whch foows from dvdg the umerator ad deomator by. 1 / ~ N(0,1), 1, ad 1 / ad the t dstrbuto. ~ ( / ) are depedet. The resut the foows from the defto of ( / )

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