Suppose that you program the screen on your friends phone to

Suppose that you program the screen on your friend\'s phone to turn on and o randomly every 2.167 seconds such that if the screen is currently on the probability it will turn o next is .7 and if the screen is currently o the probability that it will turn on next is .6.

a. Find the transition matrix P and set up the problem labeling all necessary states and variables.

b. Find the probability that after the 3rd transition the screen is o , given that the screen is currently on.

c. Find the equilibrium distribution of the system. In the long run, is the screen on or o most of the time?

Solution

Transition matrix A = 0.7 0.3

0.4 0.6

Find Eigen values and eigen vectors

The characteristic polynomial of matrix A is:

p(x)=x2?13x/10+3/10

Eigen values are = 3/10, 1 and eigen vector = (1 -4/3) and (1 1)

As spectral radius =1

The eigen vector 3/10 will extinguish in the long run.

As the eigen vector for 1 is (1 1)

more time on