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Chapter 4: Problem 35

Consider a Markov chain with states \(0,1,2,3,4\). Suppose \(P_{0,4}=1\); and suppose that when the chain is in state \(i, i>0\), the next state is equally likely to be any of the states \(0,1, \ldots, i-1 .\) Find the limiting probabilities of this Markov chain.

Short Answer

Expert verified
The limiting probabilities of this Markov Chain are: p0 = \( \frac{4}{11} \), p1 = \( \frac{3}{11} \), p2 = \( \frac{2}{11} \), p3 = \( \frac{1}{11} \), and p4 = \( \frac{1}{11} \).

Step by step solution

01

Find the transition probability matrix

The transition probabilities are given as follows: When the chain is in state 0, P(0,4) = 1. So, every time the chain is in state 0, it transitions to state 4 with probability 1. When the chain is in state i (i>0), the next state is equally likely to be any of the states from 0 to i-1. We can use this information to find the transition probabilities. Let's define the matrix P, where P(i,j) represents the probability of transitioning from state i to state j. \(P = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & 0 \\ \end{bmatrix}\)
02

Compute the limiting probabilities

To find the limiting probabilities, we want to find the probability distribution vector 蟺 such that 蟺P = 蟺, where 蟺 is a row vector with the probabilities of each state. By multiplying 蟺 with the matrix P, we get a system of linear equations representing the equilibrium condition. The sum of the elements of 蟺 should also be equal to 1, as it represents a probability distribution. We can write the system of linear equations as follows: 蟺(0) = \( \frac{1}{4} \)蟺(4) 蟺(1) = 蟺(0) + \( \frac{1}{2} \)蟺(2) + \( \frac{1}{3} \)蟺(3) + \( \frac{1}{4} \)蟺(4) 蟺(2) = \( \frac{1}{2} \)蟺(2) + \( \frac{1}{3} \)蟺(3) + \( \frac{1}{4} \)蟺(4) 蟺(3) = \( \frac{1}{3} \)蟺(3) + \( \frac{1}{4} \)蟺(4) 蟺(4) = 蟺(0) And, 蟺(0) + 蟺(1) + 蟺(2) + 蟺(3) + 蟺(4) = 1 Solving the above system of linear equations, we get the limiting probabilities: 蟺(0) = 4/11, 蟺(1) = 3/11, 蟺(2) = 2/11, 蟺(3) = 1/11, and 蟺(4) = 1/11 Hence, the limiting probabilities of this Markov Chain are: p0 = 4/11, p1 = 3/11, p2 = 2/11, p3 = 1/11, and p4 = 1/11.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Markov Chain
A Markov chain is a stochastic process that undergoes transitions from one state to another on a state space. It is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The defining characteristic of a Markov chain is that it does not depend on the previous states but solely on the current state, which means that the future is independent of the past given the present.

For example, in the exercise provided, we have a Markov chain with five states, labeled 0 through 4. Here, transitions from one state to another are determined by specific probabilities, which are quantified within a structure called a transition probability matrix.
Transition Probability Matrix
The transition probability matrix is a square matrix that describes the probabilities of moving from one state to another in a Markov chain. Each entry in the matrix, denoted as P(i,j), represents the probability of transitioning from state i to state j. The rows of this matrix correspond to the current state, and the columns correspond to the next state.

In our exercise, the transition probability matrix is constructed in a way that reflects the nature of the Markov chain: for state 0, the probability of transitioning to state 4 is 1, and for states i > 0, the probability of moving to a lower-numbered state is distributed equally among all possible lower states.
Equilibrium Condition
The equilibrium condition, also known as the steady-state condition, occurs when the probabilities of being in each state do not change over time. For a Markov chain to reach equilibrium, there exists a probability distribution vector, denoted by \( \pi \) in this exercise, which satisfies the equation \( \pi P = \pi \) when multiplied by the transition probability matrix P. This equation implies that the same probabilities result after one transition, indicating stability over time.

Finding the equilibrium distribution allows us to predict the long-term behavior of the Markov chain, which is especially valuable in various fields like economics, genetics, and computing.
System of Linear Equations
A system of linear equations is a collection of one or more linear equations involving the same set of variables. In the context of finding limiting probabilities in a Markov chain, the equilibrium condition can be expressed as a system of linear equations that must be solved to find the steady-state probabilities.

The provided exercise clearly demonstrates this - to find the limiting probabilities of the Markov chain, we construct a system where each equation represents the balance of probabilities between the states, adhering to the principles of the transition probabilities. Additionally, because we're dealing with probabilities, an extra equation that reflects the total probability summing up to 1 is included to complete the system and ensure its solvability.

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Most popular questions from this chapter

Each morning an individual leaves his house and goes for a run. He is equally likely to leave either from his front or back door. Upon leaving the house, he chooses a pair of running shoes (or goes running barefoot if there are no shoes at the door from which he departed). On his return he is equally likely to enter, and leave his running shoes, either by the front or back door. If he owns a total of \(k\) pairs of running shoes, what proportion of the time does he run barefooted?

Coin 1 comes up heads with probability \(0.6\) and coin 2 with probability \(0.5\). A coin is continually flipped until it comes up tails, at which time that coin is put aside and we start flipping the other one. (a) What proportion of flips use coin \(1 ?\) (b) If we start the process with \(\operatorname{coin} 1\) what is the probability that \(\operatorname{coin} 2\) is used ori the fifth flip? 19\. For Example 4.4, calculate the proportion of days that it rains.

For a Markov chain \(\left\\{X_{n}, n \geqslant 0\right\\}\) with transition probabilities \(P_{i, j}\), consider the conditional probability that \(X_{n}=m\) given that the chain started at time 0 in state \(i\) and has not yet entered state \(r\) by time \(n\), where \(r\) is a specified state not equal to either \(i\) or \(m\). We are interested in whether this conditional probability is equal to th茅 \(n\) stage transition probability of a Markov chain whose state space does not include state \(r\) and whose transition probabilities are $$ \underline{Q}_{i, j}=\frac{P_{i, j}}{1-P_{i, r}}, \quad i, j \neq r $$ Either prove the equality $$ P\left\\{X_{n}=m \mid X_{0}=i, X_{k} \neq r, k=1, \ldots, n\right\\}=Q_{i, m}^{n} $$ or construct a, counterexample.

At all times, an urn contains \(N\) balls - some white balls and some black balls. At each stage, a coin having probability \(p, 0

For the Markov chain with states \(1,2,3,4\) whose transition probability matrix \(\mathbf{P}\) is as specified below find \(f_{i 3}\) and \(s_{i 3}\) for \(i=1,2,3\). $$ \mathbf{P}=\left[\begin{array}{llll} 0.4 & 0.2 & 0.1 & 0.3 \\ 0.1 & 0.5 & 0.2 & 0.2 \\ 0.3 & 0.4 & 0.2 & 0.1 \\ 0 & 0 & 0 & 1 \end{array}\right] $$
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