Question

In: Statistics and Probability

The following is the transition probability matrix of a Markov chain with states 1,2,3,4 ⎛⎞ .4...

The following is the transition probability matrix of a Markov chain with states

1,2,3,4
⎛⎞

.4 .3 .2 .1 P=⎜.2 .2 .2 .4⎟ ⎝ .25 .25 .5 0 ⎠

.2 .1 .4 .3
(a) find the probability that state 3 is entered before state 4;

If X0 = 1
(b) find the mean number of transitions until either state 3 or state 4 is entered.

Solutions

Expert Solution

Answer:-

Given That:-

the transition probability matrix of a Markov chain with states 1,2,3,4

(a)the probability that state 3 is entered before state 4.

Compute

compute

The given markov chain is regular the state transition diagram for the possible transition are .

Thus,

It is indicate

It is full circle and reach any state from any

other state in less than (or) equal to 3. transitions.

The reach state is entered before state 4 means .

we reach the required state in early we just make an extra step from this state back to the same state .

So,

Here we take 3 steps.

The probability that the state 3 is entered before state 4 is

(b)

the mean number of transitions until either state 3 or state 4 is entered.

Using

P given in the problem .

gives steady state probabilities

II =

Hence,

= 3 steps

= 6 steps .

which is required solution.

orchestra answered 1 year ago

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