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Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen

Chapter 4

Stochastic Processes - all with Video Answers

Educators

Chapter Questions

00:53

Problem 1

Is a constant function, say $f(x)=c \neq 0, o(h)$ ?

Nick Johnson
Nick Johnson
Numerade Educator
01:51

Problem 2

Suppose that in New Zealand, home of the Gala apple, years for these wonderful apples can be described as great, average, or poor. Suppose that following a great year the probabilities of great, average, or poor years are $0.5,0.3$, and 0.2 , respectively. Suppose, also, that following an average year the probabilities of great, average, or poor years are $0.2,0.5$, and 0.3 , respectively. Finally, suppose that following a poor year the probabilities for great, good, or poor years are $0.2,0.2$, and 0.6 , respectively. Assume we can describe the situation from year to year by a Markov chain with the states 0,1 , and 2 corresponding to great, average, and poor years, respectively. Please do the following:
(a) Set up the transition probability matrix $P$ of this Markov chain.
(b) Suppose the initial probability for a great year is 0.2 , for an average year is $\mathbf{0 . 5}$, and for a poor year is 0.3 . Calculate the probability distribution after one year and after 5 years.

Dominador Tan
Dominador Tan
Numerade Educator
02:18

Problem 3

Consider the Markov chain with states $0,1,2,3$ with transition probability matrix
$$
P=\left[\begin{array}{llll}
0 & 0 & \frac{1}{2} & \frac{1}{2} \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0
\end{array}\right]
$$
Determine which states are transient and which are recurrent.

Nick Johnson
Nick Johnson
Numerade Educator
02:18

Problem 4

Consider the Markov chain with states $0,1,2,3,4$ with transition probability matrix
$$
P=\left[\begin{array}{ccccc}
\frac{3}{4} & \frac{1}{4} & 0 & 0 & 0 \\
\frac{3}{4} & \frac{1}{4} & 0 & 0 & 0 \\
0 & 0 & \frac{3}{4} & \frac{1}{4} & 0 \\
0 & 0 & \frac{3}{4} & \frac{1}{4} & 0 \\
\frac{1}{4} & \frac{1}{4} & 0 & 0 & \frac{1}{2}
\end{array}\right] .
$$
Determine the classes of this chain and whether each is transient or recurrent.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 5

Lucky Lily and Winning William decide to play the following game based on an urn containing nine white balls and eleven black ones.
The play proceeds as follows. A ball is drawn and replaced. If it is white, Lili wins a dollar from Winning. If the ball is black, Winning wins a dollar from Lili. Lili starts with 20 dollars and Winning with 10 dollars. The game continues until one player wins all of the other player's money. What is the probability that Winning wins?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:01

Problem 6

Two groups, Group Able and Group Baker, are competing for the same responsibility at Consolidated Craven. Group Able has a head count of 50 , that is, is authorized to have 50 people. Group Baker has a head count of 20. Each year one person is taken from one group and given to the other. If the probability that the shift is from Able to Baker is 0.52 , show that one group will disappear. Calculate the probability that Group Able will survive.

Natalie Anderson
Natalie Anderson
Numerade Educator
10:03

Problem 7

Consider Example 4.4.2. Suppose now that the probability that a 0 is received as a 1 is $\alpha$ and the probability a 1 is received as a 0 is $\beta$, so that the transition probability matrix $P$ is given by
$$
P=\left[\begin{array}{cc}
1-\alpha & \alpha \\
\beta & 1-\beta
\end{array}\right] \text {. }
$$
(a) Show that $\left(\pi_0, \pi_1\right)=(\beta /(\alpha+\beta, \alpha /(\alpha+\beta)$ is a stationary distribution.
(b) Show that $f_0^{(1)}=(1-\alpha)$ and $f_0^{(n)}=\alpha \beta(1-\beta)^{n-2}$ for $n=$ $2,3, \ldots$.
(c) Calculate the mean recurrence time $m_0=\sum_{n=1}^{\infty} n f_0^{(n)}$ and verify that $\pi_0=1 / m_0$.

Robin Corrigan
Robin Corrigan
Numerade Educator

Problem 8

Suppose $\{N(t), t \geq 0\}$ is a renewal process with renewal function $M(t)=5 t$. What is the probability distribution of the number of renewals by time 15 ?

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