Market research suggests that in a five year period 8% of people with cable television will get rid of it, and 26% of those without it will sign up for it. Compare the predictions of the Markov chain model with the following data on the fraction of people with cable TV: 56.4% in 1990, 63.4% in 1995, and 68.0% in 2000. What is the long run fraction of people with cable TV?

in the long run, 76.47% will have Cable TV

Step-by-step explanation:

Given the data in the question;

the matrix of transition from having cable TV to not having cable TV is

P = [ 0.92    0.08

       0.26    0.74  ]

Now if [ 0.564   0.436 ] is the distribution in 1990,

then in 1995 we have;

[ 0.564   0.436 ]  [ 0.92    0.08          = [ 0.6322    0.3678 ]

                              0.26    0.74  ]

so 63.22% will have cable TV in 1995

[ 0.6322    0.3678 ]  [ 0.92    0.08         = [ 0.6773    0.3227 ]

                                    0.26    0.74  ]

also 67.73% will have cable TV in 2000

let V = [ V1  V2 ] be the long run vector then

V1 + V2 = 1 lets say equ1    and  VP = V

⇒[ V1  V2 ]    [ 0.92    0.08        =  [V1  V2 ]

                       0.26    0.74  ]

⇒0.92V1 + 0.26V2 = V1

  0.08V1 + 0.74V2 = V2

OR 0.26V2 = 0.08V1

from equ 1, V1 = (1 - V2)

SO

0.26V2 = 0.08 (1 - V2)

0.26V2 = 0.08 - 0.08V2

0.34V2 = 0.08

V2 = 0.08 / 0.34

V2 = 0.2353

THUS

0.26V2 = 0.08V1

0.26(0.2353) = 0.08V1

0.061178 = 0.08V1

V1 = 0.061178 / 0.08

V1 = 0.7647

Therefore in the long run, 76.47% will have Cable TV

a is the correct

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