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STA3007
Applied probability (0809)
Solution to Assignment 4
Chapter 3
Problem 4.1
Method I
Let 1
,
2
,
3
,
4
,
5
,
6
,
7
,
8 denote the diﬀerent patterns of three tosses, HHH, HHT, HTH,
HTT, THH, THT, TTH, TTT, respectively. Then, we can get the following transition
matrix without stoping rule
P
1 =
HHH 1
/
2 1
/
2
0
0
0
0
0
0
HHT
0
0
1
/
2 1
/
2
0
0
0
0
HTH
0
0
0
0
1
/
2 1
/
2
0
0
HTT
0
0
0
0
0
0
1
/
2 1
/
2
THH 1
/
2 1
/
2
0
0
0
0
0
0
THT
0
0
1
/
2 1
/
2
0
0
0
0
TTH
0
0
0
0
1
/
2 1
/
2
0
0
TTT
0
0
0
0
0
0
1
/
2 1
/
2
.
From the above transition matrix and our intuition, we can see that all states are equally
distributed on the state space. If that HHT or HTH appears is taken as stoping rule, the
transition matrix is changed, and given as follows
P
=
HHH 1
/
2 1
/
2
0
0
0
0
0
0
HHT
0
1
0
0
0
0
0
0
HTH
0
0
1
0
0
0
0
0
HTT
0
0
0
0
0
0
1
/
2 1
/
2
THH 1
/
2 1
/
2
0
0
0
0
0
0
THT
0
0
1
/
2 1
/
2
0
0
0
0
TTH
0
0
0
0
1
/
2 1
/
2
0
0
TTT
0
0
0
0
0
0
1
/
2 1
/
2
.
Obviously, there are 2 state, HHH and THH, which can go to state HHT with probability
1
/
2, but only 1 state, THT, to state HTH with probability 1
/
2. Therefore, HHT may
appear more frequently than HTH on average.
Method II
Let 1
,
2
,
3
,
0 denote the states, H, HH, HHT and others diﬀerent from the ﬁrst three,
respectively, then we can get the following transition matrix
P
1 =
1
/
2 1
/
2
0
0
1
/
2
0
1
/
2
0
0
0
1
/
2 1
/
2
0
0
0
1
.
1
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View Full Document where state 3 is an absorption state. Let
ν
i
= E[
T

X
0
=
i
] for
i
= 0
,
1
,
2. Then by ﬁrst
step analysis, the following system equation holds
ν
0
= 1 +
1
2
ν
0
+
1
2
ν
1
ν
1
= 1 +
1
2
ν
0
+
1
2
ν
2
ν
2
= 1 +
1
2
ν
2
Solving the above system equation, we have
ν
0
= 8.
Let 1
,
2
,
3
,
0 denote the states, H, HT, HTH and others diﬀerent from the ﬁrst three,
respectively, then we can get the following transition matrix
P
1 =
1
/
2 1
/
2
0
0
0
1
/
2 1
/
2
0
1
/
2
0
0
1
/
2
0
0
0
1
.
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This note was uploaded on 02/16/2011 for the course ECON 1224 taught by Professor Afdsa during the Spring '11 term at CUHK.
 Spring '11
 afdsa

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