Question

A Markov chain \(\left\\{X_{n}, n \geqslant 0\right\\}\) with states \(0,1,2\), has the transition probability matrix $$ \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{3} & \frac{1}{6} \\ 0 & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{2} & 0 & \frac{1}{2} \end{array}\right] $$ If \(P\left\\{X_{0}=0\right\\}=P\left\\{X_{0}=1\right\\}=\frac{1}{4}\), find \(E\left[X_{3}\right]\).

Short answer

Based on the provided steps, we can calculate the expected value of the Markov chain at step 3 by following these steps: 1. Write down the initial state probabilities as \(V_0=[\frac{1}{4}, \frac{1}{4}, \frac{1}{2}]\). 2. Set up the transition matrix \(P = \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{3} & \frac{1}{6} \\\ 0 & \frac{1}{3} & \frac{2}{3} \\\ \frac{1}{2} & 0 & \frac{1}{2} \end{array}\right] \). 3. Compute the distribution after 3 steps by calculating \(V_{3} = V_{0}P^{3}\). 4. Calculate the expected value using the formula \(E[X_{3}] = \sum xP[X_{3} = x]\), where \(P[X_{3} = x]\) is the probability that variable \(X_{3}\) takes on the value x.

Step-by-step solution

Write Down the Initial State Probabilities

The initial state probabilities are given as: \(P\left\\{X_{0}=0\right\\}=\frac{1}{4}\) and \(P\left\\{X_{0}=1\right\\} = \frac{1}{4}\). Since the total probability must sum up to 1, then the probability that \(X_{0} = 2\) must be \(1 - P\left\\{X_{0}=0\right\\} - P\left\\{X_{0}=1\right\\} = \frac{1}{2}\). So we can write the initial state probabilities as a vector \(V_0=[\frac{1}{4}, \frac{1}{4}, \frac{1}{2}]\).

Set Up the Transition Matrix

The Transition matrix given in the problem is denoted as \(P\). Let's call the matrix \(P\) and represent it as: \[ P = \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{3} & \frac{1}{6} \\\ 0 & \frac{1}{3} & \frac{2}{3} \\\ \frac{1}{2} & 0 & \frac{1}{2} \end{array}\right] \]

Compute the Distribution after 3 Steps

To compute the probability distribution of the states after 3 steps, we use the formula \(V_{n}=V_{0}P^{n}\). Here, \(n = 3\), so we calculate \(V_{3} = V_{0}P^{3}\). To get \(P^{3}\), we multiply the matrix \(P\) by itself twice (i.e. squaring the matrix), then multiply this by the initial state probability vector \(V_{0}\).

Calculate the Expected Value

The expected value of a random variable is calculated as \(E[X] = \sum xP[X = x]\) where \(P[X = x]\) is the probability that variable X takes on the value x. Here, we want to find \(E[X_{3}]\), so we simply multiply each state by their respective probabilities after 3 steps (i.e. elements of the vector \(V_{3}\)), and then sum up these values to get the expected value \(E[X_{3}]\).