Stat 5102 First Midterm Exam
February 26, 2003
Name
Student ID
The exam is open book and open handouts. You may also use one 8
1
2
×
11
sheet of paper with formulas, etc. You may use a calculator, but shouldn’t
need to. Put all of your work on this test form (use the back if necessary).
Show your work or give an explanation of your answer. No credit for numbers
with no indication of where they came from.
The points for the questions total to 100. There are 2 pages and 5 prob
lems.
1. [20 pts.]
Suppose
X
1
,
X
2
,
...
,
X
n
are independent and identically
distributed with probability density function
f
(
x

θ
) =
θe

θx
,
0
< x <
∞
,
0
< θ <
∞
,
and suppose the prior distribution for
θ
has probability density function
g
(
θ
) =
β
α
Γ(
α
)
θ
α

1
e

βθ
,
0
< θ <
∞
,
where
α >
0 and
β >
0 are known numbers (the hyperparameters).
(a) Find the posterior distribution of
θ
given data
x
1
,
...
,
x
n
. You may
specify the posterior distribution either by giving its probability
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 Spring '02
 Staff
 Normal Distribution, Probability theory, probability density function

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