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STAT 428
Spring 2010
Homework 2
Feb 5
Homework 2
Due in class on Friday, Feb 12.
1.
The following is another version of the BoxMuller algorithm.
1. Generate
Y
1
and
Y
2
independently from exponential distribution with parameter 1 until
Y
2
>
(1

Y
1
)
2
/
2.
2. Generate
U
from Uniform[0
,
1] and take
X
=
b
Y
1
,
if
U
≤
0
.
5
,

Y
1
,
if
U >
0
.
5
.
(1)
Show that this algorithm produces one normal variable
X
.
2.
In the lecture on Wednesday Feb 3, we gave a rejection method for the case when the target
distribution
π
(
x
) is only known up to a normalizing constant. Show that the accepted samples
from that algorithm indeed follow the target distribution
π
(
x
).
3.
Let
X
1
,... ,X
n
be i.i.d. from
N
(
θ,
1), where
θ
is the unknown parameter. In Bayesian
inference, we may put a Cauchy prior distribution on
θ
, and the Bayes estimate of
θ
is the mean
of the posterior distribution
π
(
θ

x
1
,... ,x
n
)
∝
1
π
(1 +
θ
2
)
n
p
i
=1
1
√
2
π
e

(
x
i

θ
)
2
/
2
.
Suppose
n
= 3 and the observed values are
x
1
= 0
.
8
,x
2
= 1
.
3
,x
3
= 1
.
1. Design a rejection
algorithm to generate samples from
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This note was uploaded on 10/24/2010 for the course STAT 428 taught by Professor Chen during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Chen

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