I am struggling to get through this one. Consider an experiment with n possible states. At each step, individuals transitionfrom state i to state j with probability sij. (This sort of model is useful in computerscience, economics, statistics, and other areas.) Let S be the matrix with entries sij.This matrix satisfies: (i) all entries are ≥ 0, and (ii) all rows sum to 1. Matrices satis-fying properties (i) and (ii) are called stochastic matrices or in some books, probabilitymatrices. An example is(1).(a) For the given example S, find S² and S³, and verify that they are also stochasticmatrices.S= =(b) Let S be any n x n stochastic matrix. Explain why S1n-dimensional vector of all 1's.= 1, where 1 is the(c) Let S be any n x n stochastic matrix. Explain why SP is also stochastic, forany positive integer p. (This means part (a) was not a coincidence.) (Hint:Starting from (b), conclude S²1 = 1 by writing S2 = SS and performing themultiplications in a suitable order. Repeat the reasoning p times to concludeSP1 = 1.)(d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SPmean?

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Answered: Consider an experiment with n possible…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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