斯托克、沃森着《计量经济学》(第二版)答案

V2 Y
E[(Y

P Y
)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
4 Stock/Watson - Introduction to Econometrics - Second Edition
probability
Y0 0.25
Y1 0.50
Y2 0.25
(b) Cumulative probability distribution function for Y
Outcome (number of heads)
Probability
Y0 0
0dY1 1dY2
0.25
0.75
Yt2 1.0
2. We know from Table 2.2 that Pr (Y 0) 022, Pr (Y 1) 078, Pr ( X 0) 030, Pr ( X 1) 070. So
(a)
P Y来自百度文库
E(Y) 0 u Pr (Y
0) 1u Pr (Y
1)
0 u 022 1u 078 078,
Solutions to Exercises in Chapter 2 5
To compute the kurtosis, use the formula from exercise 2.21: E(X P)4 E(X4 ) 4[E(X)][E(X3)] 6[E(X)]2[E(X2)] 3[E(X)]4 0.3 4 u 0.32 6 u 0.33 3u 0.34 0.0777
030 u (078) u 007 030 u 022 u 063
0084,
V
cor (X , Y )
XY
VV
XY
0084
04425
021u 01716
3. For the two new random variables W 3 6X and V 20 7Y, we have:
Alternatively, E( X P)4 = [(1 0.3)4 u 0.3] [(0 0.3)4 u 0.7] 0.0777 Thus, kurtosis is E(X P)4/V 4 = .0777/0.464 1.76
(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
(a) E(V) E(20 7Y) 20 7E(Y) 20 7u 078 1454, E(W) E(3 6X) 3 6E(X) 3 6 u 070 72
(b)
V2 W
var (3 6X)
62

V
2 X
36 u 021 756,
V2 V
var (20 7Y)
(c)
P Y
=
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
E(Y 2 ) (02 u 0.25) (12 u 0.50) (22 u 0.25) 1.50 so that var(Y) E(Y 2 ) [E(Y)]2 1.50 (1.00)2 0.50.
P X
E(X)
0 u Pr (X
0) 1u Pr (X 1)
0 u 030 1u 070 070
(b)
V2 X
E[(X PX )2 ]
(0 0.70)2 u Pr (X 0) (1 0.70)2 u Pr (X 1)
(070)2 u 030 0302 u 070 021
(c) Table 2.2 shows Pr (X 0, Y 0) 015, Pr (X 0, Y 1) 015, Pr (X 1, Y 0) 007, Pr (X 1, Y 1) 063. So
V XY
cov (X , Y )
E[( X

P X
)(Y

P Y
)]
(0 - 0.70)(0 - 0.78) Pr( X 0, Y 0)
(7)2

V
2 Y
49u 01716
84084
(c)
V WV
cor (W , V )
cov (3 6X , 20 7Y ) 6(7) cov (X , Y )
V WV
VV WV
3528 756 u 84084
04425
42 u 0084
3528
4. (a) E( X 3 ) 03 u (1 p) 13 u p p (b) E( X k ) 0k u (1 p) 1k u p p (c) E( X ) 0.3 var ( X ) E( X 2 ) [E( X )]2 0.3 0.09 0.21 Thus, V 0.21 0.46.
To compute the skewness, use the formula from exercise 2.21: E(X P)3 E(X3) 3[E(X2 )][E(X)] 2[E(X)]3 0.3 3u 0.32 2 u 0.33 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 E(X P)3/V 3 .084/0.463 0.87.
PART ONE Solutions to Exercises
Chapter 2
Review of Probability
Solutions to Exercises
1. (a) Probability distribution function for Y
Outcome (number of heads)