Question

On any given day a receptionist is either cheerful (c) or gloomy (g). Suppose that whether or not the receptionist will be cheerful tomorrow depends on her mood only through the last two days. Specifically, suppose that if she was cheerful yesterday and today, then she will be cheerful tomorrow with probability 0.8; if she was cheerful yesterday but not today then she will be cheerful tomorrow with probability 0.3; if she is cheerful today but not yesterday then she will be cheerful tomorrow with probability 0.4; and if she was gloomy yesterday and today then she will be cheerful tomorrow with probability 0.2.

(i) Compute the one-step transition probability matrix of the above Markov chain. (Hint: This is a four state M.C.)

(ii) What proportion of days is the receptionist cheerful? gloomy?

Answer 1

The one-step transition probability matrix of the Markov chain can be calculated and the proportion of cheerful and gloomy days can be determined.

Step-by-step explanation:

The given problem can be represented as a four-state Markov chain. We can define the states as follows: state 1 represents the receptionist being cheerful for two consecutive days, state 2 represents the receptionist being cheerful yesterday but not today, state 3 represents the receptionist being cheerful today but not yesterday, and state 4 represents the receptionist being gloomy for two consecutive days.

The one-step transition probability matrix can be calculated as follows:

P = [[0.8, 0, 0, 0], [0.3, 0, 0, 0], [0, 0.4, 0, 0], [0, 0, 0.2, 0]]

To find the proportion of days the receptionist is cheerful, we need to sum up the probabilities of being in states 1, 2, and 3. So, the proportion of cheerful days is 0.8 + 0.3 + 0.4 = 1.5.

The proportion of gloomy days can be found by subtracting the proportion of cheerful days from 1. So, the proportion of gloomy days is 1 - 1.5 = -0.5.