11 Suppose that some other comic book villain has programmed

11. Suppose that some other comic book villain has programmed a light to randomly switch between off and on independently every second for some nefarious reason. the light is programmed such that if it is currently on it has a 70% chance of turning off next, and if it is currently off it has a 50% chance of turning on next. Suppose that on is represented as state 1 and off is represented as state 2. (a) What is the transition matrix associated with this system? (b) What is the long run stationary distribution of this system?

Solution

The transition matrix associated with the question is:

For a long run stationary state,

Consider a matrix P (1 x 2) such that:

P * T = P

Thus,

0.7a + 0.3b = a

0.5a + 0.5b = b

Solving these equations simultaneously,

we get

0.7a = 0.7b

i.e a = b

And thus.

Also, we know that a + b = 1

Thus, a = b = 0.5 will be the long run stationary distribution.

Hope this helps.

Current State	----->	1	2
Next State
1		0.3	0.5
2		0.7	0.5