Question

Consider a simplification of a autonomous vehicle project on Markov processes. The process has three states that describe the driving environment: s₁:= “driving in city,” s₂ := “driving in suburbs,” s₃ := “driving in rural area (‘country’).” If the car is driving in the city, there is a 50% chance the next passenger pickup assignment will also be in the city. If the car is driving in the suburbs, there is a 40% chance the next passenger pickup assignment will be in the suburbs. If the car is driving in the country, there is a 20% chance the next passenger pickup assignment will also be in the country. Anytime there is a switch from the driving environment, it is equally likely to be to either of the other two environments.

1. If the car starts in the city, what is the probability that the tenth passenger pickup will be in the country?

2. Suppose a typical drive in the city results in asset depreciation (loss from wear-and-tear) of $10; for the suburbs depreciation is $5, and for the country it is $3. What is the expected loss from depreciation after the 100’th passenger?

Answer 1

STATIONARY DISTRIBUTION:

Stationary distribution may refer to:

• A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assuming irreducibility, the stationary distribution is always unique if it exists, and its existence can be implied by positive recurrence of all states. The stationary distribution has the interpretation of the limiting distribution when the chain is ergodic.
• The marginal distribution of a stationary process or stationary time series
• The set of joint probability distributions of a stationary process or stationary time series.