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
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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
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10
a.
From the given information,
Consider,
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.
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^3 is also stochastic matrix.
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