Problem # 5. Let X (n) denote a Markov chain with states 0 and 1. The transition probability matrix is given by P ( } 1/2 1/2 1/4 3/4 which means, for example, that if the system is in state 1 it will transition to state 0 with probability 1/4. a) Score some easy points by finding the steady-state probability of state 0. b) If the process starts in state 0 at n = 0, (X(0) will be in state 1 after 2 time units, i.e., at n = 2? 0), what is the probability that it c) Suppose instead that the probability of starting at state 0 is 1/5, (P[X(0) = 0] = 1/5). At time n = 2, you observe that the system is in state 1, (X(2) = 1). Given this observation, what is now the probability that the system actually started at X(0) = 0?
Problem # 5. Let X (n) denote a Markov chain with states 0 and 1. The transition probability matrix is given by P ( } 1/2 1/2 1/4 3/4 which means, for example, that if the system is in state 1 it will transition to state 0 with probability 1/4. a) Score some easy points by finding the steady-state probability of state 0. b) If the process starts in state 0 at n = 0, (X(0) will be in state 1 after 2 time units, i.e., at n = 2? 0), what is the probability that it c) Suppose instead that the probability of starting at state 0 is 1/5, (P[X(0) = 0] = 1/5). At time n = 2, you observe that the system is in state 1, (X(2) = 1). Given this observation, what is now the probability that the system actually started at X(0) = 0?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Problem # 5.
Let X (n) denote a Markov chain with states 0 and 1. The transition
probability matrix is given by
P
( }
1/2 1/2
1/4 3/4
which means, for example, that if the system is in state 1 it will transition to state 0 with
probability 1/4.
a) Score some easy points by finding the steady-state probability of state 0.
b) If the process starts in state 0 at n = 0, (X(0)
will be in state 1 after 2 time units, i.e., at n = 2?
0), what is the probability that it
c) Suppose instead that the probability of starting at state 0 is 1/5, (P[X(0) = 0] = 1/5).
At time n = 2, you observe that the system is in state 1, (X(2) = 1). Given this
observation, what is now the probability that the system actually started at X(0) = 0?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad306ca9-2ffb-49a4-9b86-39f2ab9633a1%2F358bc32b-aba9-4249-8f07-e2b75c297835%2Ftdusq2k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem # 5.
Let X (n) denote a Markov chain with states 0 and 1. The transition
probability matrix is given by
P
( }
1/2 1/2
1/4 3/4
which means, for example, that if the system is in state 1 it will transition to state 0 with
probability 1/4.
a) Score some easy points by finding the steady-state probability of state 0.
b) If the process starts in state 0 at n = 0, (X(0)
will be in state 1 after 2 time units, i.e., at n = 2?
0), what is the probability that it
c) Suppose instead that the probability of starting at state 0 is 1/5, (P[X(0) = 0] = 1/5).
At time n = 2, you observe that the system is in state 1, (X(2) = 1). Given this
observation, what is now the probability that the system actually started at X(0) = 0?
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