Question

linear algebra

Another use of the transition matrix is in stochastic modeling. An example is the following: Imagine a park with three locations: a lake; a picnic area; and a playground. Every hour, on the hour, the parkgoers move according to the following rulcs: Half of those at the lake move to the picnic area, and one-quarter of those at the lake move to the playground. Half of those at the picnic area go to the lake, and the other half go to the playground. Half of those at the playground go to the picnic area. Once they arrive, the parkgoers stay until the next hour, at which point they move again, according to the same rules. (a) Suppose there are 100 persons at each location. How many persons are there at each of the locations life, it's often easier to determine where you've come from than it is to determine where you're going to. hour later? Suggestion: In one (b) Let p, n. n be variables representing the number of persons at the picnic area, lake, and playground at hour equations for pn+1, n+1 , gn+1 , giving the number of persons at the picnic area, lake, and playground at hour n + 1. Write down n. (c) Write down the coefficient matrix T for the system of equations you just wrote. (The first, second, and third columns should cor- respond to the variables pn, n, and gn, respectively; and the first. sccond, and third rows should correspond to the cquations giving Pu+1, In+1, and gn+1, respectively). (d) The matrix you've produced is called (since it is for a of the following matrices could be a cannot be. Identify the matrix that cannot be a stochastic matrix. and explain why it can't be one. Also identify the model that matrix that can be a stochastic matrix, and give the corresponding transition matrix or a stochastic matrix. One stochastic matrix: the other stochastic model) a movement rules. (1/4 0 1/3 1/2 1/4 1/2V A 1/2 1/4 1/2. B 1/2 1/2 1/6 0 1/2 1/4, 1/4 1/2 1/2/ (c) Gencralize your obscrvation: A matrix M can be the stochastic matrix for a model provided the properties of a mensions and entries. Defend your conclusion by connecting the requirement to a feature of the stochastic model) (Give stochastic matrix, including its possible di-

Answer 1

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