ECON 506M (solution)
Quiz3A, November 13, 2007
NAME:
1.
X and Y have a joint bivariate distribution.
f(x,y) = 3x
if 0 < y < x < 1
0
otherwise.
For a and b below, state the distribution, state the limits, and draw a small chart of the
distribution.
a.
[8]
Find the marginal distribution of X.
g(x) =
dy
x
x
∫
0
3
= 3x
2
for 0<x< 1
=
0
elsewhere
b. [10]
Find the conditional distribution of (Y  X = 0.6)
h (YX = 0.6) = [3x / 3x
2
] (for x= 0.6) = 1.66
for 0<y<x
= 0
elsewhere
c.
[12]
Find E(XY)
E(XY) =
3
.
0
)
2
.
0
5
.
0
)
(
3
)
3
(
1
0
5
2
1
0
4
1
0
1
2
1
0
1
=

=

=
=
∫
∫∫
∫∫
y
y
dy
y
y
ydxdy
x
dxdy
x
xy
y
y
2. [10]
Let
)
,
(
~
1
1
p
n
Bin
W
,
)
,
(
~
2
2
p
n
Bin
W
and the two random variables are
independent. Use MGF method to find the mass function for
2
1
W
W
W
+
=
.
2
1
*
)
1
(
1
*
)
(
*
)
(
)
(
2
1
n
n
t
t
W
W
W
e
p
e
p
t
M
t
M
t
M
+


=
=
)
,
(
~
2
1
p
n
n
Bin
W
+
3.
An economist is examining individual incomes in her country where it is assumed that
incomes are normally distributed. The variance of individual income in the population is
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 Fall '08
 Staff
 Probability theory, true population, 1 2 j, joint bivariate distribution., countries currency

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