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Chapter 4: Problem 63

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 \(i=1,2,3\). $$ \mathbf{P}=\left[\begin{array}{llll} 0.4 & 0.2 & 0.1 & 0.3 \\ 0.1 & 0.5 & 0.2 & 0.2 \\ 0.3 & 0.4 & 0.2 & 0.1 \\ 0 & 0 & 0 & 1 \end{array}\right] $$

Short Answer

Expert verified
In summary, to find the first passage probabilities \(f_{i3}\) and mean first passage times \(s_{i3}\) for \(i=1,2,3\) with given transition probability matrix \(\mathbf{P}\), we utilize the recursive formula for first passage probabilities and the definition of mean first passage time. Calculation involves iterating over the recursions until a desired level of precision is obtained.

Step by step solution

01

Calculate the first passage probabilities, \(f_{i3}\)

The first passage probabilities \(f_{ij}\) are defined as the probability that the Markov chain will first visit state \(j\) from state \(i\). They can be calculated recursively using the following formula: $$ f_{ij}^{(n)} = \sum_{k \neq j} p_{ik} f_{kj}^{(n-1)}, \quad n \geq 1, $$ where \(f_{ij}^{(n)}\) denotes the probability that the process first enters state \(j\) at step \(n\) and \(p_{ik}\) is the transition probability from state \(i\) to state \(k\). For \(i=1,2,3\), we will compute \(f_{i3}\) using the formula above: 1. When \(i = 1\), \(f_{13}^{(n)} = 0.1\cdot f_{33}^{(n-1)}+0.2\cdot f_{23}^{(n-1)}+0.4\cdot f_{13}^{(n-1)}\). 2. When \(i = 2\), \(f_{23}^{(n)} = 0.2\cdot f_{33}^{(n-1)}+0.5\cdot f_{23}^{(n-1)}+0.1\cdot f_{13}^{(n-1)}\). 3. When \(i = 3\), \(f_{33}^{(n)} = 0.2\cdot f_{33}^{(n-1)}+0.4\cdot f_{23}^{(n-1)}+0.3\cdot f_{13}^{(n-1)}\). With these recursions, we can calculate the first passage probabilities until the desired convergence is obtained.
02

Calculate the mean first passage times, \(s_{i3}\)

The mean first passage time, \(s_{ij}\), is the expected time it takes for the Markov chain to reach state \(j\) for the first time from state \(i\). It can be calculated using the first passage probabilities, \(f_{ij}^{(n)}\), as follows: $$ s_{ij} = \sum_{n=1}^{\infty} n \cdot f_{ij}^{(n)}. $$ Using the previously calculated first passage probabilities, \(f_{ij}^{(n)}\), for \(i=1,2,3\), we can now compute the mean first passage times, \(s_{i3}\), by summing the series for each \(i\). Remember that, in general, we must use numerical methods to compute both first passage probabilities and mean first passage times. By iterating over the recursions, we can obtain these values to a desired level of precision.

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Key Concepts

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

Transition Probability Matrix
Imagine you are playing a board game with a multi-sided die. Each roll determines your next move. The transition probability matrix in a Markov chain is akin to the die, which decisively shapes the game's progression.

At its core, the transition probability matrix, often denoted as \( \mathbf{P} \), encapsulates all the chances of moving from one state to another in a single step within a Markov process. It's a square matrix where each \( p_{ij} \) element signifies the probability of the system transitioning from state \( i \) to state \( j \). Importantly, the probabilities in each row must add up to 1 since from any given state, the transition must go to some state in the next step.

To better understand and utilize this matrix:
  • Verify that it is square and that the rows add up to one.
  • When interpreting the matrix, consider the row as the current state and the column as the potential next state.
  • Use this matrix to predict long-term behavior by raising it to a high power, which provides insights into the steady-state distribution.
Mean First Passage Time
In the context of Markov chains, the mean first passage time (MFPT) is much like the average time it takes for a newcomer to find a specific location in a maze. This average time, denoted as \( s_{ij} \), gives a measure of how long it takes, on average, to reach state \( j \) for the first time starting from state \( i \).

Calculating MFPT is a matter of using the first passage probabilities \( f_{ij}^{(n)} \) for each step \( n \), meaning, the probability that state \( j \) is visited for the first time exactly at step \( n \). By summing the products of these probabilities with their corresponding step number \( n \) across all steps, we obtain \( s_{ij} \)鈥攁 weighted sum that tells us the expected number of steps to reach state \( j \) from state \( i \).

To compute this effectively:
  • Ensure you have the accurate first passage probabilities. Inaccurate probabilities will yield incorrect MFPTs.
  • Be prepared for a potentially infinite series summation, which may require numerical methods for practical computation.
  • Understand that MFPT has both theoretical and practical implications, such as optimizing strategies in games or predicting system behavior in real-world scenarios.
Recursive Probability Calculation
The concept of recursive probability calculation can be likened to a family tree, where to understand an individual's ancestry, one must recursively trace back through the generations. Similarly, in recursive calculation for Markov chains, the probability of transitioning to a specific state is determined through an iterative process that builds upon previously computed probabilities.

With a recursive approach, the first passage probabilities \( f_{ij}^{(n)} \) at step \( n \) are computed by summing the probabilities of moving to some intermediary state \( k \) and then, from \( k \) to target state \( j \) in one less step (\( n-1 \)). This technique leverages the property that Markov chains have no memory鈥攆uture states depend only on the current state, not the path taken to reach it.

Key points to consider when applying recursive probability calculation:
  • The process iteratively builds upon the transition probabilities calculated in previous steps, requiring careful tracking at each iteration.
  • Initial conditions are vital for kicking off the recursion; without them, the calculation cannot proceed.
  • Stopping criteria must be defined to truncate the process once probabilities stabilize, ensuring practical computation time and resources.

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Most popular questions from this chapter

Let the transition probability matrix of a two-state Markov chain be given, as in Example 4.2, by $$ \mathbf{P}=\left\|\begin{array}{cc} p & 1-p \\ 1-p & p \end{array}\right\| $$ Show by mathematical induction that $$ \mathbf{P}^{(n)}=\left\|\begin{array}{ll} \frac{1}{2}+\frac{1}{2}(2 p-1)^{n} & \frac{1}{2}-\frac{1}{2}(2 p-1)^{n} \\ \frac{1}{2}-\frac{1}{2}(2 p-1)^{n} & \frac{1}{2}+\frac{1}{2}(2 p-1)^{n} \end{array}\right\| $$

A transition probability matrix \(\mathbf{P}\) is said to be doubly stochastic if the sum over each column equals one; that is, $$ \sum_{i} P_{i j}=1, \quad \text { for all } j $$ If such a chain is irreducible and aperiodic and consists of \(M+1\) states \(0,1, \ldots, M\), show that the limiting probabilities are given by $$ \pi_{j}=\frac{1}{M+1}, \quad j=0,1, \ldots, M $$

A particle moves on a circle through points which have been mark茅d \(0,1,2,3,4\) (in a clockwise order). At each step it has a probability \(p\) of moving to the right (clockwise) and \(1-p\) to the left (counterclockwise). Let \(X_{n}\) denote its location on the circle after the \(n\) th step. The process \(\left[X_{n}, n \geqslant 0\right\\}\) is a Markov chain. (a) Find the transition probability matrix. (b) Calculate the limiting probabilities.

Let \(\pi_{i}\) denote the long-run proportion of time a given Markov chain is in state \(i\). (a) Explain why \(\pi_{i}\) is also the proportion of transitions that are into state \(i\) as well as being the proportion of transitions that are from state \(i\). (b) \(\pi_{i} P_{i j}\) represents the proportion of transitions that satisfy what property? (c) \(\sum_{i} \pi_{i} P_{i j}\) represent the proportion of transitions that satisfy what property? (d) Using the preceding explain why $$ \pi_{j}=\sum_{i} \pi_{i} P_{i j} $$

Trials are performed in sequence. If the last two trials were successes, then the next trial is a success with probability \(0.8\); otherwise the next trial is a success with probability 0.5. In the long run, what proportion of trials are successes?
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