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Statistics 150 (Stochastic Processes): Midterm Exam, Spring 2009. J. Pitman, U.C. Berkeley.
1. A sequence of random variables
X
1
,X
2
,...
, each with two possible values 0 and 1, is such that
P
(
X
1
= 1) =
p
1
and for each
n
≥
1
P
(
X
n
+1
= 1

X
1
,...,X
n
) = (1

θ
n
)
p
1
+
θ
n
S
n
/n
where (
θ
n
) is a sequence of parameters with 0
≤
θ
n
≤
1, and
S
n
:=
X
1
+
···
+
X
n
. Find and prove a
formula for
P
(
X
n
= 1) in terms of
p
1
and
θ
1
,θ
2
,...
.
2. Consider a Markov chain (
X
n
) with transition matrix
P
. For 1
≤
m < n
ﬁnd and explain a formula
for the conditional distribution of
X
m
given
X
0
=
i
and
X
n
=
k
in terms of the entries of appropriate
matrix powers of
P
.
3. Consider a
p
↑
,
1

p
↓
walk (
S
n
) started at
S
0
=
a
and run until the time
T
when it ﬁrst hits either 0
or
b
for some positive integers 0
≤
a
≤
b
. Assuming
p
6
= 1
/
2, justify an application of Wald’s identity
to derive a simple formula for
E
(
T

S
0
=
a
) in terms of
p
and the known solution of the gambler’s ruin
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This note was uploaded on 10/02/2009 for the course STAT 87528 taught by Professor Pitman,jim during the Spring '09 term at Berkeley.
 Spring '09
 Pitman,Jim
 Statistics

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