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Stat 5101
Stat 5101 (Geyer) Midterm 2
Problem 1
This is a multivariate change of variable problem (Section 12.1 in Lindgren).
The first step is to solve for the old variables in terms of the new variables. Clearly
X
=
U V
.
Then
Y
=
U

X
=
U
(1 
V
).
The next step is to determine the Jacobian of the transformation
Clearly
U
=
X Y
can be any positive number. Given
U
,
can be any number between 0 and 1 because
X
can be any number between
X
and
U
. Thus the joint density is
Problem 2
This is a job for the ``recognizing the unnormalized density'' trick. First
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If we recognize the integrand as a
density, we see the integral must give
No further simplification is possible. The recursion formula for gamma functions only allows cancellation of
gamma functions with arguments differing by an integer.
Problem 3
(a)
Let
X
be the random variable in question. The count of points occurring in a fixed time interval of length
t
for
a Poisson process with rate parameter
is
. Here
per day and
t
= 7 days. Thus the answer is
.
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 Spring '02
 Staff
 Normal Distribution, Probability theory, Poisson process, Yi, Bernoulli random variables, P. D. F.

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