individual has three umbrellas, some at her office, and some at home 1.38. An 1.

Question

individual has three umbrellas, some at her office, and some at home 1.38. An 1.38. aving home in the morning (or leaving work at night) and it is raining, If she is If she is ill take an umbrella, if one is there. Otherwise, she gets wet. Assume that nt of the past, it rains on each trip with probability 0.2. To formulate a indebou chain, let X, be the number of umbrellas at her current location. (a) Find ition probability for this Markov chain. (b) Calculate the limiting fraction she of time she gets wet

Explanation / Answer

Taking into account the shift in view point, i.e., if we have 0 umbrellas at the current location,

we will always have 3 at the next current location, one can compute p(i, j) as

0 1 2 3

0 0 0 0 1

1 0 0 0.8 0.2

2 0 0.8 0.2 0

3 0.8 0.2 0 0

The stationary distribution is (.2105, .2631, .2631, .2631)

This means that getting wet is (.2)(.2105) = 0.0421.

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