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1、PART ONE Solutions to Exercises Chapter 2 Review of Probability ? ? Solutions to Exercises 1. (a) Probability distribution function for Y Outcome (number of heads) Y ? 0 Y ? 1 Y ? 2 probability 0.25 0.50 0.25 (b) Cumulative probability distribution function for Y Outcome (number of heads) Y ? 0 0 ?
2、Y ? 1 1 ? Y ? 2 Y ? 2 Probability 0 0.25 0.75 1.0 (c) = ( )(00.25)(1 0.50)(2 0.25)1.00 Y E Y? Using Key Concept 2.3: 22 var( )() ( ) ,YE YE Y? and ? 2222 ()(00.25)(10.50)(20.25)1.50E Y so that 222 var( )() ( )1.50(1.00)0.50.YE YE Y? 2. We know from Table 2.2 that ? ?Pr(0)0 22,Y ? ?Pr(1)0 78,Y ? ?Pr(
3、0)0 30,X ? ?Pr(1)0 70.X So (a) ( )0Pr(0)1 Pr(1) 00 221 0 780 78, ( )0Pr(0)1 Pr(1) 00 301 0 700 70 Y X E YYY E XXX ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (b) 22 22 22 22 22 22 () (00.70)Pr(0)(10.70)Pr(1) ( 0 70)0 300 300 700 21 () (00.78)Pr(0)(10.78)Pr(1) ( 0 78)0 220 220 780 1716 XX YY E X XX E Y YY ? ? ?
4、? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 4 Stock/Watson - Introduction to Econometrics - Second Edition (c) Table 2.2 shows Pr(0,0)0 15,XY? ? Pr(0,1)0 15,XY? ? Pr(1,0)0 07,XY? ? Pr(1,1)0 63.XY? ? So cov( ,)()() (0-0.70)(0-0.78)Pr(0,0) (00 70)(10 78)Pr(01) (10 70)(00 78)Pr(10) (10 70)(10 78)Pr(11) ( 0 70)(
5、0 78)0 15( 0 70)0 220 15 0 30( 0 78)0 070 XYXY X YE XY XY XY XY XY ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?300 220 63 0 084, 0 084 ( ,)0 4425 0 21 0 1716 XY XY cor X Y ? ? ? ? ? ? ? ? ? ? ? ? ? 3. For the two new random variables 36WX? and 207 ,VY? we have: (a) ( )(207 )207 ( )207
6、 0 7814 54, ()(36 )36 ( )36 0 707 2 E VEYE Y E WEXE X ? ? ? ? ? ? ? ? (b) 222 222 var(36 )636 0 217 56, var(207 )( 7)49 0 17168 4084 WX VY X Y ? ? ? ? ? ? ? ? ? (c) (36, 207 )6( 7)( ,)420 0843 528 3 528 (,)0 4425 7 56 8 4084 WV WV WV covXYcov X Y cor W V ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 4. (a) ? 333
7、()0(1)1E Xppp (b) ?()0(1)1 kkk E Xppp (c) ?()0.3E X ? 22 var()() ()0.30.090.21XE XE X Thus, ?0.210.46.? To compute the skewness, use the formula from exercise 2.21: 3323 23 ()()3 () ( )2 ( ) 0.33 0.32 0.30.084 E XE XE XE XE X? ? ? ? Alternatively, ? 333 () = (10.3)0.3(00.3)0.70.084E X? Thus, skewnes
8、s 333 () /.084/0.460.87.E X? Solutions to Exercises in Chapter 2 5 To compute the kurtosis, use the formula from exercise 2.21: 443224 234 ()()4 ( ) ()6 ( ) ()3 ( ) 0.34 0.36 0.33 0.30.0777 E XE XE XE XE XE XE X? ? ? ? ? Alternatively, ? 444 () = (10.3)0.3(00.3)0.70.0777E X? Thus, kurtosis is 444 ()
9、 /= .0777/0.461.76E X? ? 5. Let X denote temperature in ?F and Y denote temperature in ?C. Recall that Y ? 0 when X ? 32 and Y ? 100 when X ? 212; this implies (100/180) (32) or 17.78(5/9).YXYX? ? Using Key Concept 2.3, ?70 F X ? implies that 17.78(5/9) 7021.11 C, Y ? ? and ? ?7 F X ? implies(5/9) 7
10、3.89 C. Y ? 6. The table shows that Pr(0,0)0 045,XY? ? Pr(0,1)0 709,XY? ? Pr(1,0)0 005,XY? ? Pr(1,1)0 241,XY? ? Pr(0)0 754,X? ? ? ?Pr(1)0 246,X ? ?Pr(0)0 050,Y ? ?Pr(1)0 950.Y (a) ? ? ? ? ? ? ? ( )0Pr(0)1 Pr(1) 00 0501 0 9500 950 Y E YYY? (b) (unemployed) Unemployment Rate (labor force) Pr(0)0 05010
11、 9501( ) # # YE Y ? ? ? ? ? ? (c) Calculate the conditional probabilities first: Pr(0,0)0 045 Pr(0|0)0 0597, Pr(0)0 754 XY YX X ? ? ? ? Pr(0,1)0 709 Pr(1|0)0 9403, Pr(0)0 754 XY YX X ? ? ? ? Pr(1,0)0 005 Pr(0|1)0 0203, Pr(1)0 246 XY YX X ? ? ? ? Pr(1,1)0 241 Pr(1|1)0 9797 Pr(1)0 246 XY YX X ? ? ? ?
12、The conditional expectations are ( |1)0Pr(0|1)1 Pr(1|1) 00 0203 1 0 97970 9797, ( |0)0Pr(0|0)1 Pr(1|0) 00 05971 0 94030 9403 E Y XYXYX E Y XYXYX ? ? ? ? ? ? ? ? ? ? ? ? ? ? 6 Stock/Watson - Introduction to Econometrics - Second Edition (d) Use the solution to part (b), Unemployment rate for college
13、grads 1( |1)10.97970.0203. Unemployment rate for non-collegegrads 1( |0)10.94030.0597. E Y X E Y X ? ? ? ? ? ? (e) The probability that a randomly selected worker who is reported being unemployed is a college graduate is Pr(1,0)0 005 Pr(1|0)0 1 Pr(0)0 050 XY XY Y ? ? ? ? ? The probability that this
14、worker is a non-college graduate is Pr(0|0)1Pr(1|0)10 10 9XYXY? ? ? ? ? ? ? (f) Educational achievement and employment status are not independent because they do not satisfy that, for all values of x and y, Pr(|)Pr()YyXxYy? For example, Pr(0|0)0 0597Pr(0)0 050YXY? ? ? 7. Using obvious notation, ?;CM
15、F thus ? CMF ? and 222 2cov(,). CMF M F? This implies (a) ?4045$85,000 C ?per year. (b) (,) (,), MF Cov M F cor M F ? ? ? so that (,)(,). MF Cov M Fcor M F? ? Thus (,)12 18 0.80172.80,Cov M F ? where the units are squared thousands of dollars per year. (c) 222 2cov(,), CMF M F? so that 222 12182 172.80813.60, C ? ? and ?813.6028.524 C ? thousand d
