Question

Assume that the probability of rain tomorrow is 0.7 if it is raining today, and assume that the probability of its being clear (no rain) tomorro w is 0.9 if it is clear today. Also assume that these probabilities do not change if information is also provided about the weather before today.

a) Explain why the stated assumptions imply that the Markovian property holds for the evolution of the weather.

b) Formulate the evolution of the weather as a Markov chain by identifying the random variables of interest and defining the possible states of these random variables

c) Write an expression for evaluating the random variables

d) Determine its one - step transition m atrix

Answer 1

a) The Markov assumption states that the future value of a process should be determined by the present state alone with no dependency on the past values. Here since today's weather determines whether it will rain tomorrow or be clear and the probability do not change if we know the state of an earlier day, we can see that the Markov assumption holds here.

B) The random variables of interest are whether it will rain tomorrow or be clear tomorrow. We have two states Rain, Clear depending on Rain or Clear today.

The one step transition matrix P is as follows: (Column represent future state and Row represent current state) Each row sums to 1.

R C

R . 0.7 0.3

C. 0.1 0.9