Homework Help is Here – Start Your Trial Now!
Math
Statistics
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 ( (a) For the given example S, find S² and S3, and verify that they are also stochastic S= 2

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 ( (a) For the given example S, find S² and S3, and verify that they are also stochastic S= 2

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter10: Matrices
Section10.2: Addition And Subtraction Of Matrices
Problem 43E
See similar textbooks
icon
Related questions
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
Transcribed Image Text: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
steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS