Prof. Bryan Caplan
bcaplan@gmu.edu
http://www3.gmu.edu/departments/economics/bcaplan
Economics 345
Fall, 1998
Problem Set #1 Answer Key
Wallace and Silver Problems.
1.1.
a. Since the problem specifies that there is no replacement, you have to be careful when calculating the probability of different events. The best approach is to make a table (or tree) showing all possibilities and their probabilities:
1st draw |
2nd draw |
3rd draw |
Prob |
O |
Y |
Y |
(3/5)(2/4)(1/3)=6/60 |
Y |
O |
Y |
(2/5)(3/4)(1/3)=6/60 |
Y |
O |
Y |
(2/5)(1/4)(3/3)=6/60 |
Y |
O |
O |
(2/5)(3/4)(2/3)=12/60 |
O |
Y |
O |
(3/5)(2/4)(2/3)=12/60 |
O |
O |
Y |
(3/5)(2/4)(2/3)=12/60 |
O |
O |
O |
(3/5)(2/4)(1/3)=6/60 |
Reading off the table, P(X=1)=18/60; P(X=2)=36/60; P(X=3)=6/60;
P(Y=0)=6/60; P(Y=1)=36/60; P(Y=2)=18/60
b.
Y/X |
1 |
2 |
3 |
P(Y) |
0 |
0 |
0 |
6/60 |
6/60 |
1 |
0 |
36/60 |
0 |
36/60 |
2 |
18/60 |
0 |
0 |
18/60 |
P(X) |
18/60 |
36/60 |
6/60 |
c. E(X)=1.80; E(Y)=1.20; Var(X)=.36; Var(Y)=.36
d. Cov(X,Y)=-.36
e. P(1)=54/60; P(2)=0; P(3)=6/60
Z/X |
1 |
2 |
3 |
P(Z) |
1 |
18/60 |
36/60 |
0 |
54/60 |
2 |
0 |
0 |
0 |
0 |
3 |
0 |
0 |
6/60 |
6/60 |
P(X) |
18/60 |
36/60 |
6/60 |
f. If X=1, then Y=2, so |X-Y|=|1-2|=1 with probability 1.
g. E(Z)=1.2; V(Z)=.360; Cov(X,Y)=.24
1.8.
a. 8
b. V(X-Y)=Var(X)+Var(Y)+2*Cov(X,Y)=6
c. 44
d. (a) 8; (b) 0; (c) 80
1.9. X must be a constant.
1.16. X has a normal distribution with mean of 66 and SD=10 (Since Var(X)=100). Define Z=(X-66/10)~N(0,1). Thus P(X<50) equals P(Z<(50-66)/10)=.0548.
Freedman and Lane Problems.
p.30, #5.
a. T. Mathematically, , so it can't be negative. Intuitively, you can have a big spread, a small spread, or no spread, but you can't have negative spread!
b. T
c. F
d. T
e. F
p.35, #1.
a. T
b. F
c. T
d. F
p.46, #7.
a. n
b.
c.
d.
p.53, #4.
a. 0
b. 0
c. 1
d. 1
e. T. By definition, . So plugging in for (a)-(d), .
Caplan Problem.
a. ;
b. Use t-distribution for 4 degrees of freedom. Critical value is 2.776; , so reject the null.
Computer Work.
Variable |
Obs |
Mean |
Var |
SD |
Min |
Max |
I |
35 |
4.84 |
9.61 |
3.10 |
1 |
13.5 |
U |
35 |
6.14 |
2.27 |
1.51 |
3.5 |
9.7 |
I |
U |
|
I |
1.000 |
.207 |
U |
.207 |
1.000 |
Also, you can download the completed workfile from the class webpage.