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A particle moves on a circle through points which been marked 0,1,2,3,4 (in a clockwise order). At each step it has a probability p of moving to the right (clockwise) and 1- p to the left (counterclockwise). Let Xn denote its location on the circle after nth step. The process { Xn, n>= 0} is a Markov Chain a. Find the transition probability matrix. b. Calculate the limiting probabilities.

Respuesta :

Answer:

Step-by-step explanation:

Given data:

SS={0,1,2,3,4}

Let probability of moving to the right be = P

Then probability of moving to the left is =1-P

The transition probability matrix is:

[tex]\left[\begin{array}{ccccc}1&P&0&0&0\\1-P&1&P&0&0\\0&1-P&1&P&0\\0&0&1-P&1&P\\0&0&0&1-P&1\end{array}\right][/tex]

Calculating the limiting probabilities:

π0=π0+Pπ1                 eq(1)

π1=(1-P)π0+π1+Pπ2     eq(2)

π2=(1-P)π1+π2+Pπ3    eq(3)

π3=(1-P)π2+π3+Pπ4    eq(4)

π4=(1-P)π3+π4             eq(5)

Ï€0+Ï€1+Ï€2+Ï€3+Ï€4=1

Ï€0-Ï€0-PÏ€1=0

→π1 = 0

substituting value of π1  in eq(2)

(1-P)Ï€0+PÏ€2=0

from

π2=(1-P)π1+π2+Pπ3  

we get

(1-P)Ï€1+PÏ€3 = 0

from

Ï€3=(1-P)Ï€2+Ï€3+PÏ€4

we get

(1-P)Ï€2+PÏ€4 =0

from π4=(1-P)π3+π4  

→π3=0

substituting values of π1 and π3 in eq(3)

→π2=0

Now

Ï€0+Ï€1+Ï€2+Ï€3+Ï€4=0

Ï€0+Ï€4=1

Ï€0=0.5

Ï€4=0.5

So limiting probabilities are {0.5,0,0,0,0.5}

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