Question

Experience has shown that the general mood of Henry may be realistically modeled as a three-state Markov process with the mutually exclusive collectively exhaustive states: State 1: Cheerful State 2: So-so State 3: Glum His mood can change only overnight and the following transition probabilities apply to each night’s trial:            0.0 0.3 0.7 0.3 0.4 0.3 0.6 0.2 0.2 P 1 2 3 1 2 3 Problem 10 Three balls are divided between two containers. During each period a ball is randomly chosen and switched to the other container. (a) Find (in the steady state) the fraction of the time that a container will contain 0, 1, 2 or 3 balls. (b) If container 1 contains no balls, on the average how many periods will go by before it again contains no balls? (Note: This is a special case of the Ehrenfest Diffusion model, which is used in biology to model diffusion through a membrane) We are told that there is a 50-50% chance that Henry’s mood today (Monday) is so-so or cheerful. (a) Determine the probability distribution for Henry’s mood this Thursday and a week from today. (b) Determine this probability distribution for a day a few months hence. Is the answer dependent on the initial conditions? Based on these results, what would you say about Henry’s mood? is he a cheerful or more of a gloomy type of person? (c) We are told that Henry’s mood today is so-so, using the results from the first passage times, can we expect him to become glum before he becomes cheerful? (d) If he is glum today, when is the earliest we can expect him to be cheerful?

Answer 1