Question

In: Advanced Math

Consider the following Markov chain with P{X0 = 2} = 0.6 and P{X0 = 4} =...

Consider the following Markov chain with P{X0 = 2} = 0.6 and P{X0 = 4} = 0.4:

1 2 3 4 5 6

1 0 0 0 0 1 0

2 .2 .05 0 .6 0 .15

3 0 0 .8 0 0 .2

4 0 .6 0 .2 0 .2

5 1 0 0 0 0 0

6 0 0 .7 0 0 .3

a. What is P{X1 = 4, X2 = 6 | X0 = 2}?

b. What is P{X2 = 6 | X0 = 2}? What is P{X18 = 6 | X16 = 2}?

c. What is P{X0 = 2, X1 = 4, X2 = 6}?

d. What is P{X1 = 4, X2 = 6}?

Expert Solution

The matrix is

Starting at , the distribution of is

and the distribution of is

Also, given , the distribution of is

Thus, , and . Now, by law of total probability, and Markovian property,

b) As found above, . By Markovian property, we have

c) We have

d) .

Colby Messinger answered 1 year ago

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