Find a transition Matrix

Question

If the student attends class on a certain Friday, then he is five times as likely to be absent the next Friday as to attend. If the student is absent on a certain Friday, then he is twice as likely to attend class the next Friday as to be absent again.

Assume that state 1 is Attends Class and that state 2 is Absent from Class.

(Note: Express your answers as rational fractions or as decimal fractions rounded to 4 decimal places (if the answers have more than 4 decimal places).)

(1) Find the transition matrix for this Markov process.

 

 

(2) Find the three-step transition matrix P(3)

 

 

(3) If the student is absent on a given Friday afternoon, what is the probability that he will be present 3 weeks later?

 

 

Answer 1

Hi Katie,

As the state 1 and 2 are to attend the class and to be absent, respectively,
a transition matrix, T, for this Markov process has its elements as follows:

     t₁₁ = p,   t₂₁ = 5p
     t₁₂ = 2q,  t₂₂ = q
    
As t₁₁ + t₂₁ = 1 and t₁₂ + t₂₂ = 1,

     t₁₁ = 1/6,  t₂₁ = 5/6
     t₁₂ = 2/3,  t₂₂ = 1/3

(1) T = ⌈ t₁₁   t₁₂ ⌉ = ⌈ 1/6   2/3 ⌉
            ⌊ t₂₁   t₂₂ ⌋    ⌊ 5/6   1/3 ⌋

(2) P(3) = T³

 

             = T·T·T

             = ⌈ 3/8   1/2 ⌉
                 ⌊ 5/8   1/2 ⌋

(3) The answer is the 1st row, 2nd column element of P(3),
     which is 1/2.

 

Hope these help.