Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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 satis- fying properties (i) and (ii) are called stochastic matrices or in some books, probability matrices. An example is 1120 s-(11). S= (a) For the given example S, find S² and S3, and verify that they are also stochastic matrices. (b) Let S be any n x n stochastic matrix. Explain why S1 = 1, where 1 is the n-dimensional vector of all 1's. (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.) (d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SP mean?

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 satis- fying properties (i) and (ii) are called stochastic matrices or in some books, probability matrices. An example is 1120 s-(11). S= (a) For the given example S, find S² and S3, and verify that they are also stochastic matrices. (b) Let S be any n x n stochastic matrix. Explain why S1 = 1, where 1 is the n-dimensional vector of all 1's. (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.) (d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SP mean?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

I am struggling to get through this one.

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 satis- fying properties (i) and (ii) are called stochastic matrices or in some books, probability matrices. An example is (1). (a) For the given example S, find S² and S³, and verify that they are also stochastic matrices. S= = (b) Let S be any n x n stochastic matrix. Explain why S1 n-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, 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.) (d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SP mean?
Transcribed Image Text: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 satis- fying properties (i) and (ii) are called stochastic matrices or in some books, probability matrices. An example is (1). (a) For the given example S, find S² and S³, and verify that they are also stochastic matrices. S= = (b) Let S be any n x n stochastic matrix. Explain why S1 n-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, 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.) (d) Interpret SP in terms of the experiment, i.e. what does the (i, j)th entry of SP mean?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 7 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS