(c) XY cov (X , Y ) E[( X X )(Y Y )]
(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) (078) 015 (070) 022 015
030 (078) 007 030 022 063
0084,
XY 0084
corr (X , Y ) 04425
XY 021 01716
2.3. For the two new random variables W 3 6 X and V 20 7Y , we have:
(a) E (V ) E (20 7Y ) 20 7 E (Y ) 20 7 078 1454,
E (W ) E (3 6 X ) 3 6 E ( X ) 3 6 070 72
(b) W2 var (3 6 X ) 62 X2 36 021 756,
V2 var (20 7Y ) (7)2 Y2 49 01716 84084
(c) WV cov(3 6 X , 20 7Y ) 6 (7)cov(X , Y ) 42 0084 3528
WV 3528
corr (W , V ) 04425
WV 756 84084
2.4. (a) E ( X 3 ) 03 (1 p) 13 p p
(b) E ( X k ) 0k (1 p) 1k p p
(c) E ( X ) 0.3 , and var(X) = E(X2)−[E(X)]2 = 0.3 −0.09 = 0.21. Thus = 0.21 = 0.46.
var ( X ) E ( X ) [ E ( X )] 0.3 0.09 0.21 0.21 0.46. To compute the skewness, use
2 2
the formula from exercise 2.21:
E ( X )3 E ( X 3 ) 3[ E ( X 2 )][ E ( X )] 2[ E ( X )]3
0.3 3 0.32 2 0.33 0.084
Alternatively, E ( X )3 [(1 0.3)3 0.3] [(0 0.3)3 0.7] 0.084
Thus, skewness E ( X )3/ 3 0.084/0.463 0.87.
To compute the kurtosis, use the formula from exercise 2.21:
E ( X ) 4 E ( X 4 ) 4[ E ( X )][ E ( X 3 )] 6[ E ( X )]2 [ E ( X 2 )] 3[ E ( X )]4
0.3 4 0.32 6 0.33 3 0.34 0.0777
Alternatively, E ( X )4 [(1 0.3)4 0.3] [(0 0.3)4 0.7] 0.0777
Thus, kurtosis is E ( X )4/ 4 0.0777/0.464 1.76
, 4 Stock/Watson • Introduction to Econometrics, Third Edition
2.5. 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 Y (100/180) ( X 32) or Y 17.78 (5/9) X. Using Key
Concept 2.3, X 70oF implies that Y 17.78 (5/9) 70 21.11C, and X 7oF implies
Y (5/9) 7 3.89C.
2.6. The table shows that Pr ( X 0, Y 0) 0037, Pr ( X 0, Y 1) 0622,
Pr ( X 1, Y 0) 0009, Pr ( X 1, Y 1) 0332, Pr ( X 0) 0659, Pr ( X 1) 0341,
Pr (Y 0) 0046, Pr (Y 1) 0954.
(a) E (Y ) Y 0 Pr(Y 0) 1 Pr (Y 1)
0 0046 1 0954 0954
# (unemployed)
(b) Unemployment Rate
# (labor force)
Pr (Y 0) 1 Pr(Y 1) 1 E (Y ) 1 0954 0.046
(c) Calculate the conditional probabilities first:
Pr ( X 0, Y 0) 0037
Pr (Y 0| X 0) 0056,
Pr ( X 0) 0659
Pr ( X 0, Y 1) 0622
Pr (Y 1| X 0) 0944,
Pr ( X 0) 0659
Pr ( X 1, Y 0) 0009
Pr (Y 0| X 1) 0026,
Pr ( X 1) 0341
Pr ( X 1, Y 1) 0332
Pr (Y 1| X 1) 0974
Pr ( X 1) 0341
The conditional expectations are
E (Y |X 1) 0 Pr (Y 0| X 1) 1 Pr (Y 1| X 1)
0 0026 1 0974 0974,
E (Y |X 0) 0 Pr (Y 0| X 0) 1 Pr (Y 1|X 0)
0 0056 1 0944 0944
(d) Use the solution to part (b),
Unemployment rate for college graduates 1 E(Y|X 1) 1 0.974 0.026
Unemployment rate for non-college graduates 1 E(Y|X 0) 1 0.944 0.056
(e) The probability that a randomly selected worker who is reported being unemployed is a
college graduate is
Pr ( X 1, Y 0) 0009
Pr ( X 1|Y 0) 0196
Pr (Y 0) 0046
The probability that this worker is a non-college graduate is
Pr ( X 0|Y 0) 1 Pr ( X 1|Y 0) 1 0196 0804
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