 1 
University of Colorado
Department of Economics
Economics 3818
Midterm Examination
Prof. Jeffrey S. Zax
3 November 2005
Solutions
(32)
1.
X and Y have the following joint distribution:
Y
y
1
=2
y
2
=4
x
1
=2
.2
.1
Xx
2
=4
.1
.1
x
3
=6
.2
.3
(1)
a.
What is P(X=4, Y=4)?
P(X=4, Y=4)=.1.
(2)
b.
What is the marginal distribution of X?
P(X=2)=.3, P(X=4)=.2, P(X=6)=.5.
(3)
c.
What is P(Y=4X=4)?
()
( )
PY
X
X
PX
X
==
=
=∩ =
=
=
=
44
4
4
1
2
5

,
.
.
..
(4)
d.
Is P(Y=4)=P(Y=4X=4)? Why or why not? What, if anything, does the relation
ship between these two probabilities tell you with regard to the question of
whether X and Y are independent?
.5=P(Y=4)=P(Y=4X=4)=.5. This means that X and Y could be independent.
But it does not prove that they are. A proof would require demonstrating that
this relationship was true for all values of Y and X.
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(2)
e.
What is the conditional distribution of X given that Y=2?
P(Y=2) =
P(Y=2,X=2)+P(Y=2,X=4)+P(Y=2,X=6)
=
.2
+.1 +.2 =.5. Therefore,
()
( )
PX
Y
Y
PY
Y
Y
Y
Y
==
=
=
=
=
=
=
22
2
2
5
4
42
2
1
5
2
62
2
2
5
4

,
.
.
.,

,
.
.

,
.
.
..
(3)
f.
What is E(X)?
Using the answer to b.,
() () ()
EX
xpx
ii
i
=
=++=
=
∑
1
3
23 42 65 44
...
.
.
(2)
g.
What is E(Y)?
Using the answer to e.,
P(Y=4) =
P(Y=4,X=2)+P(Y=4,X=4)+P(Y=4,X=6)
=
.1
+.1 +.3 =.5. Therefore,
() ()
EY
ypy
i
=
=+=
=
∑
1
2
25 45 3
.
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 Fall '05
 JeffreyS.Zax
 Economics, Variance, Probability theory, probability density function

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