Solution Manual for Stochastic Processes With R An Introduction 1st edition

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SOLUTIONS MANUAL

, TABLE OF CONTENTS
CHAPTER 1 ……………………………………………………………………………………. 3

CHAPTER 2 ……………………………………………………………………………………. 31

CHAPTER 3 ……………………………………………………………………………………. 41

CHAPTER 4 ……………………………………………………………………………………. 48

CHAPTER 5 ……………………………………………………………………………………. 60

CHAPTER 6 ……………………………………………………………………………………. 67

CHAPTER 7 ……………………………………………………………………………………. 74

CHAPTER 8 ……………………………………………………………………………………. 81

CHAPTER 9 ……………………………………………………………………………………. 87




2

, CHAPTER
1 0.3 0.4 0.3
EXERCISE 1.1. For a Markov chain with a one-step transition probability matrix � 0.2 0.3 0.5
�
0.8 0.1 0.1
we compute:

(a) 𝑃𝑃(𝑋𝑋3 = 2 |𝑋𝑋0 = 1, 𝑋𝑋1 = 2, 𝑋𝑋2 = 3) = 𝑃𝑃(𝑋𝑋3 = 2 | 𝑋𝑋2 = 3) (by the Markov property)
= 𝑃𝑃32 = 0.1.
(b) 𝑃𝑃(𝑋𝑋4 = 3 |𝑋𝑋0 = 2, 𝑋𝑋3 = 1) = 𝑃𝑃(𝑋𝑋4 = 3 | 𝑋𝑋3 = 1) (by the Markov property)
= 𝑃𝑃13 = 0.3.
(c) 𝑃𝑃(𝑋𝑋0 = 1, 𝑋𝑋1 = 2, 𝑋𝑋2 = 3, 𝑋𝑋3 = 1) = 𝑃𝑃(𝑋𝑋3 = 1 | 𝑋𝑋0 = 1, 𝑋𝑋1 = 2, 𝑋𝑋2 = 3) 𝑃𝑃(𝑋𝑋2 = 3 |𝑋𝑋0 = 1,
𝑋𝑋1 = 2) 𝑃𝑃(𝑋𝑋1 = 2 | 𝑋𝑋0 = 1) 𝑃𝑃(𝑋𝑋0 = 1) (by conditioning)
= 𝑃𝑃(𝑋𝑋3 = 1 | 𝑋𝑋2 = 3) 𝑃𝑃(𝑋𝑋2 = 3 | 𝑋𝑋1 = 2) 𝑃𝑃(𝑋𝑋1 = 2 | 𝑋𝑋0 = 1) 𝑃𝑃(𝑋𝑋0 = 1) (by the Markov property)

= 𝑃𝑃31 𝑃𝑃23 𝑃𝑃12 𝑃𝑃(𝑋𝑋0 = 1) = (0.8)(0.5)(0.4)(1) = 0.16.
(d) We first compute the two-step transition probability matrix. We obtain

0.3 0.4 0.3 0.3 0.4 0.3 0.41 0.27 0.32
(2)
𝐏𝐏 = � 0.2 0.3 0.5 � � 0.2 0.3 0.5 � = � 0.52 0.22 0.26�.

Now we 0.8 0.1 0.1 0.8 0.1 0.1 0.34 0.36 0.30
write
𝑃𝑃(𝑋𝑋0 = 1, 𝑋𝑋1 = 2, 𝑋𝑋3 = 3, 𝑋𝑋5 = 1) = 𝑃𝑃(𝑋𝑋5 = 1 | 𝑋𝑋0 = 1, 𝑋𝑋1 = 2, 𝑋𝑋3 = 3) 𝑃𝑃(𝑋𝑋3 = 3 |𝑋𝑋0 = 1,
𝑋𝑋1 = 2) 𝑃𝑃(𝑋𝑋1 = 2 | 𝑋𝑋0 = 1) 𝑃𝑃(𝑋𝑋0 = 1) (by conditioning)
= 𝑃𝑃(𝑋𝑋5 = 1 | 𝑋𝑋3 = 3) 𝑃𝑃(𝑋𝑋3 = 3 | 𝑋𝑋1 = 2) 𝑃𝑃(𝑋𝑋1 = 2 | 𝑋𝑋0 = 1) 𝑃𝑃(𝑋𝑋0 = 1) (by the Markov property)
(0.34)(0.26)(0.4)(1) = 0.03536.
𝑃𝑃 𝑃𝑃(𝑋𝑋 = 1) =
(2) (2)


= 𝑃𝑃31 𝑃𝑃23 12 0

EXERCISE 1.2. (a) We plot a diagram of the Markov chain.

#specifying transition probability matrix
tm<- matrix(c(1, 0, 0, 0, 0, 0.5, 0, 0, 0, 0.5, 0.2, 0, 0, 0, 0.8,
0, 0, 1, 0, 0, 0, 0, 0, 1, 0), nrow=5, ncol=5, byrow=TRUE)

#transposing transition probability matrix tm.tr<- t(tm)

#plotting diagram library(diagram)
plotmat(tm.tr, arr.length=0.25, arr.width=0.1, box.col="light blue", box.lwd=1, box.prop=0.5,
box.size=0.12, box.type="circle", cex.txt=0.8, lwd=1, self.cex=0.3, self.shiftx=0.01, self.shifty=0.09)




3

, State 2 is reflective. The chain leaves that state in one step. Therefore, it forms a separate
transient class that has an infinite period.

Finally, states 3, 4, and 5 communicate and thus belong to the same class. The chain can
return to either state in this class in 3, 6, 9, etc. steps, thus the period is equal to 3. Since
there is a positive probability to leave this class, it is transient.

The R output supports these findings.

#creating Markov chain object library(markovchain)
mc<- new("markovchain", transitionMatrix=tm,states=c("1", "2", "3", "4", "5"))

#computing Markov chain characteristics recurrentClasses(mc)

"1"

transientClasses(mc)

"2"

"3" "4" "5"

absorbingStates(mc)

"1"
4