Question

Problem 3

Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the probability of its being clear (no rain) tomorrow 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) Find the n-step transition matrix for n = 3, 7, 10, 20.

b) The probability of rain today is 0.5 Use the results in the part (a) to determine the probability that it rains in n days, for n = 3, 7, 10, 20.

c) Formulate the steady-state equations to obtain the steady state probabilities for this process. Compare the probabilities of the transition matrices of n steps of part (a) with these of the steady state as n grows.

Answer 1

(C)

for steady state probablity we get P(rain on nth day)=0.1667

P(no rain on nth day)=0.8333

Form nth step matrix we get P(X_3=1)=.2807

P(X_3=0)=0.5023*.5=0.25115