斯托克、沃森着《计量经济学》(第二版)答案

0) 007,
V XY
cov (X , Y )
E[( X P X )(Y PY )]
(0 - 0.70)(0 - 0.78) Pr( X 0, Y 0) (0 070)(1 078) Pr ( X 0 Y 1) (1 070)(0 078) Pr ( X 1 Y 0) (1 070)(1 078) Pr ( X 1 Y 1) (070) u (078) u 015 (070) u 022 u 015 030 u (078) u 007 030 u 022 u 063 0084,
0.25
0.50
0.25
(b) Cumulative probability distribution function for Y Outcome (number of heads) Probability Y0 0 0dY1 0.25 1dY2 0.75 Yt2 1.0
(c) PY = E (Y ) (0 u 0.25) (1u 0.50) (2 u 0.25) 1.00 Using Key Concept 2.3: var(Y ) E (Y 2 ) [E (Y )]2 , and
1.50 (1.00)2
0.50.
0) 022, Pr (Y
1) 078, Pr ( X
0) 030,
PY
E(Y ) 0 u Pr (Y
0) 1u Pr (Y 1)
0 u 022 1u 078 078,
PX
(b)
2 VX
E( X ) 0 u Pr ( X
0) 1u Pr ( X 1)
PART ONE Solutions to Exercises
Chapter 2
Review of Probability

1.
Solutions to Exercises
(a) Probability distribution function for Y Outcome (number of heads) probability Y 0 Y 1 Y 2
4
Stock/Watson - Introduction to Econometrics - Second Edition
(c) Table 2.2 shows Pr ( X 0, Y Pr ( X 1, Y 1) 063. So
0) 015, Pr ( X
0, Y 1) 015, Pr ( X 1, Y
E (Y 2 ) (0 2 u 0.25) (12 u 0.50) (2 2 u 0.25) 1.50
so that var(Y ) E(Y 2 ) [E (Y )]2 2. We know from Table 2.2 that Pr (Y Pr ( X 1) 070. So (a)
0.21 0.46.
0.3 0.09 0.21
To compute the skewness, use the formula from exercise 2.21: E ( X P )3 E ( X 3 ) 3[ E( X 2 )][E ( X )] 2[E ( X )]3 0.3 3 u 0.32 2 u 0.33
0 u 030 1u 070 070
E[( X P X )2 ] (0 0.70)2 u Pr ( X 0) (1 0.70)2 u Pr ( X 1) (070)2 u 030 030 2 u 070 021
2 VY
E[(Y PY )2 ] (0 0.78)2 u Pr (Y 0) (1 0.78)2 u Pr (Y 1) (078)2 u 022 0222 u 078 01716
5
To compute the kurtosis, use the formula from exercise 2.21: E ( X P )4 E( X 4 ) 4[E ( X )][E( X 3 )] 6[E( X )]2 [ E( X 2 )] 3[ E( X )]4 0.3 4 u 0.32 6 u 0.33 3 u 0.34 Thus, kurtosis is E ( X P )4/V 4 = .0777/0.464 1.76 5. Let X denote temperature in qF and Y denote temperature in qC. Recall that Y 0 when X 32 and Y 100 when X 212; this implies Y (100/180) u ( X 32) or Y 17.78 (5/9) u X. Using Key Concept 2.3, P X 70qF implies that PY 17.78 (5/9) u 70 21.11qC, and V X 7qF implies V Y (5/9) u 7 3.89qC.
cor (X , Y ) 3.
V XY V XVY
0084 021 u 01716
04425
For the two new random variables W (a)
3 6 X and V
Байду номын сангаас
20 7Y , we have:
E (V ) E (20 7Y ) 20 7E (Y ) 20 7 u 078 1454, E (W ) E (3 6 X ) 3 6E ( X ) 3 6 u 070 72 (b)
E ( X P )3/V 3 .084/0.463
0.084
Alternatively, E ( X P )3 = [(1 0.3)3 u 0.3] [(0 0.3)3 u 0.7] 0.084 Thus, skewness
0.87.
Solutions to Exercises in Chapter 2
2 VW 2 var (3 6 X ) 62 V X 2
36 u 021 756, 49 u 01716 84084
V
(c)
2 V
2 var (20 7Y ) (7) V Y
V WV
cor (W , V ) 4.
cov (3 6 X , 20 7Y ) 6(7) cov (X , Y ) 42 u 0084 3528
V WV VWVV
3528 756 u 84084 p p
04425
(a) E ( X 3 ) 03 u (1 p) 13 u p (b) E ( X ) 0 u (1 p ) 1 u p
k k k
(c) E ( X ) 0.3 var ( X ) Thus, V E ( X 2 ) [ E ( X )]2