Question

In Example 4.3, Gary is in a cheerful mood today. Find the expected number of days until he has been glum for three consecutive days.

Short answer

The expected number of days until Gary has been glum for three consecutive days can be calculated using a Markov chain with four states representing Gary's mood. Define the transition probabilities, and then solve the system of linear equations formed by these probabilities: \(E(0) = 1 + C * E(0) + G * E(1), E(1) = 1 + C * E(0) + G * E(2),\) and \(E(2) = 1 + C * E(0)\). \(E(0)\) is the expected number of days to reach state 3 (three consecutive glum days) from state 0 (cheerful mood today). Note that the cheerful (C) and glum (G) probabilities need to be provided to find the numerical answer.

Step-by-step solution

Define the states of the Markov chain

We will define four states for the Markov chain, depending on Gary's mood: State 0: Gary is in a cheerful mood (today). State 1: Gary has been glum for one day. State 2: Gary has been glum for two consecutive days. State 3: Gary has been glum for three consecutive days (our target). These states will help us understand the transitions between different moods and calculate the expected number of days.

Define the transition probabilities of the Markov chain

Let the transition probability between states i and j be denoted as P(i,j), with i and j representing the different states defined in Step 1. Since Gary's mood can only change daily, we will assume that the probability of being glum tomorrow (G) and being cheerful tomorrow (C) remain constant. The transition probabilities for the Markov chain: P(0,0) = C (cheerful mood today and cheerful tomorrow), P(0,1) = G (cheerful mood today and glum tomorrow), P(1,0) = C (one glum day and cheerful tomorrow), P(1,1) = 0 (we can't have only one glum day, as there must be at least two), P(1,2) = G (one glum day, and glum tomorrow, making it two consecutive glum days), P(2,0) = C (two consecutive glum days, and cheerful tomorrow), P(2,1) = 0 (we can't have only one glum day, as there must be at least two), P(2,2) = 0 (we can't have only two glum days, again), P(2,3) = G (two consecutive glum days, and glum tomorrow, making it three consecutive glum days).

Calculate the expected number of days

Let's denote the expected number of days to reach state 3 (three consecutive glum days) from state i as E(i). Using the transition probabilities, we can define the following recursive equations: E(0) = 1 + C * E(0) + G * E(1), E(1) = 1 + C * E(0) + G * E(2), E(2) = 1 + C * E(0) + G * E(3). Since E(3) = 0 (as we have already reached the target state), we can rewrite the equations as: E(0) = 1 + C * E(0) + G * E(1), E(1) = 1 + C * E(0) + G * E(2), E(2) = 1 + C * E(0). Solving this system of linear equations will give us the expected number of days to reach state 3 (three consecutive glum days) from state 0 (cheerful mood today). Note that we would need the values for C and G probabilities to find a numerical answer. Once you have the values for these probabilities, you can solve the equations to find the expected number of days until Gary has been glum for three consecutive days.