Question

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?

Answer 1

Answer: in the long run, 76.47% will have Cable TV

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