100000 1 110. 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. Matricessatisfying properties (i) and (ii) are called stochastic matrices. An example isS =(0(a) For the given example S, find S² and S³, and verify that they are also stochasticmatrices.Explain why S1 = 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.)=(b) Let S be any n x n stochastic matrix.n-dimensional vector of all 1's.22(d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SPmean?MacBook Air

Answered: Consider an experiment with n possible…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

1
0
000
0 1 1
10. Consider an experiment with n possible states. At each step, individuals transition
from state i to state j with probability sij. (This sort of model is useful in computer
science, 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
satisfying properties (i) and (ii) are called stochastic matrices. An example is
S =
(
0
(a) For the given example S, find S² and S³, and verify that they are also stochastic
matrices.
Explain why S1 = 1, where 1 is the
(c) Let S be any n x n stochastic matrix. Explain why SP is also stochastic, for
any positive integer p. (This means part (a) was not a coincidence.) (Hint:
Starting from (b), conclude S²1 1 by writing S² = SS and performing the
multiplications in a suitable order. Repeat the reasoning p times to conclude
SP1 = 1.)
=
(b) Let S be any n x n stochastic matrix.
n-dimensional vector of all 1's.
2
2
(d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SP
mean?
MacBook Air

Expert Solution

Step 1

Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts. In case you require the unanswered parts also, kindly re-post that parts separately.

From the given information,

$S = (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 1 \end{matrix})$

Consider,

$S^{2} = S * S = (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 1 \end{matrix}) * (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 1 \end{matrix}) = (\begin{matrix} \frac{1}{4} & \frac{1}{4} + \frac{1}{2} \\ 0 & 1 \end{matrix}) = (\begin{matrix} \frac{1}{4} & \frac{3}{4} \\ 0 & 1 \end{matrix})$

Here, all the entries are >=0 and the sum of the row elements is equal to 1.

That is, the conditions for the stochastic matrix are satisfied.

Hence, the matrix S^2 is also stochastic matrix.

$S^{3} = S^{2} * S = (\begin{matrix} \frac{1}{4} & \frac{3}{4} \\ 0 & 1 \end{matrix}) * (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 1 \end{matrix}) = (\begin{matrix} \frac{1}{8} & \frac{1}{8} + \frac{3}{4} \\ 0 & 1 \end{matrix}) = (\begin{matrix} \frac{1}{8} & \frac{7}{8} \\ 0 & 1 \end{matrix})$