11. Suppose that some other comic book villain has programmed a light to randoml

Question

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?

Explanation / Answer

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

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