AEB 6933
–
Mathematical Statistics for Food and Resource Economics
Examination I, September 29, 2006
1.
Given
,
f
x y
such that
2
2
3
1
,
10
352
4
f
x y
x
y
xy
defined over
,
2,2
x y
.
a.
Is this a valid probability density function? (10 points)
b.
Derive the marginal distribution of
y
. (10 points)
c.
Derive the conditional distribution of
x
given
y
. (10 points)
d.
Are
x
and
y
independent? (10 points)
2.
Given that
x
is distributed
0,5
U
and that
2
4
y
x
x
for
0,5
x
(and
100,0
y
):
a.
What are the conditions required for deriving
g
y
(the probability density
function for
y
) based on
y
x
? (10 points)
b.
Derive
g
y
. (10 points)
3.
Given the distribution function
2
3
1
2
f
x
x
defined over the interval
0,1
x
a.
Compute the first four population moments of the distribution. (10 points)
b.
Compare these moments with the sample moments in Table 1, do you think
that this sample was drawn from this population? (10 point)
4.
Given that
1
2
2
3
1
~
,
1
1
5
x
x
N
x
a.
Compute the conditional expectation of
2
x
such that
1
0
x
. (10 points)
b.
Compute the conditional variance of
2
x
. (10 points)
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Table 1. Sample Data for Question 3
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 Fall '09
 CARRIKER
 Probability theory, probability density function

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