Question

Consider the Markov chain with state space {0, 1, 2, ...} and transition probabilities p(m, m + 1) = 1/2 (1 - 1 / (m + 2)) for m >= 0, p(m, m - 1) = 1/2 (1 + 1 / (m + 2)) for m >= 1, and p(0, 0) = 1 - p(0, 1) = 3/4. Find the stationary distribution pi. (Hint: Note that this is a birth and death chain and use the observation that sum_{i=0}^{n} 3 / ((i + 1)(i + 3)) = 3/2 sum_{i=0}^{n} (1 / (i + 1) - 1 / (i + 3)) = 3/2 (1 + 1/2 - 1 / (n + 2) - 1 / (n + 3)) for any n >= 0.)

          Consider the Markov chain with state space {0, 1, 2, ...} and transition probabilities

p(m, m + 1) = 1/2 (1 - 1 / (m + 2)) for m >= 0,
p(m, m - 1) = 1/2 (1 + 1 / (m + 2)) for m >= 1,

and p(0, 0) = 1 - p(0, 1) = 3/4. Find the stationary distribution pi. (Hint: Note that this is a birth and death chain and use the observation that

sum_{i=0}^{n} 3 / ((i + 1)(i + 3)) = 3/2 sum_{i=0}^{n} (1 / (i + 1) - 1 / (i + 3)) = 3/2 (1 + 1/2 - 1 / (n + 2) - 1 / (n + 3))

for any n >= 0.)

Consider the Markov chain with state space 0, 1, 2, ... and transition probabilities

p(m, m + 1) = 1/2 (1 - 1 / (m + 2)) for m >= 0,
p(m, m - 1) = 1/2 (1 + 1 / (m + 2)) for m >= 1,

and p(0, 0) = 1 - p(0, 1) = 3/4. Find the stationary distribution pi. (Hint: Note that this is a birth and death chain and use the observation that

sumi=0^n 3 / ((i + 1)(i + 3)) = 3/2 sumi=0^n (1 / (i + 1) - 1 / (i + 3)) = 3/2 (1 + 1/2 - 1 / (n + 2) - 1 / (n + 3))

for any n >= 0.)

Added by Donna S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

02/23/2023

Step 1

The stationary distribution satisfies the following balance equations: $$ \pi(m)p(m, m+1) = \pi(m+1)p(m+1, m) $$ For m > 0, we have: $$ \pi(m)\frac{2m+2}{(m+1)(m+3)} = \pi(m+1)\frac{2}{1+m+2} $$ Rearranging the terms, we get: $$ \pi(m+1) = Show more…

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Consider the Markov chain with state space {0, 1, 2, ...} and transition probabilities p(m, m + 1) = 1/2 (1 - 1/(m + 2)) for m >= 0, p(m, m - 1) = 1/2 (1 + 1/(m + 2)) for m >= 1, and p(0, 0) = 1 - p(0, 1) = 3/4. Find the stationary distribution pi. (Hint: Note that this is a birth and death chain and use the observation that sum_{i=0}^{n} 3/((i + 1)(i + 3)) = 3/2 sum_{i=0}^{n} (1/(i + 1) - 1/(i + 3)) = 3/2 (1 + 1/2 - 1/(n + 2) - 1/(n + 3)) for any n >= 0.)