time-homogeneous stochastic process (Xn : n ≥ 0)
where Xn, the state of the weather on day n, has the
value 1 if the weather is sunny, or the value 2 if the weather is
rainy. For each n ≥ 1, Xn+1, given
(Xn, Xn-1), is conditionally independent of
Hn-2 = {X0, . . . , Xn-2}.
The conditional distribution of Xn+1 given the two
most recent states of the process is as follows:
- if it was sunny both yesterday and today, then it will be
sunny tomorrow with probability 0.9;
- if it was rainy yesterday but sunny today, then it will be
sunny tomorrow with probability 0.8;
- if it was sunny yesterday but rainy today, then it will be
sunny tomorrow with probability 0.7;
- if it was rainy both yesterday and today, then it will be
sunny tomorrow with probability 0.6.
- The Weather At A Holiday Resort Is Modelled As A Time Homogeneous Stochastic Process Xn N 0 Where Xn The State Of 1 (64.14 KiB) Viewed 59 times
Define today's state of the system, Yn, in terms of the pair (Xn, Xn-1) as follows: = Yn = 1, if (Xn, Xn-1) = (1,1) 2, if (Xn, Xn-1) = (1,2) 3, if (Xn, Xn-1) (2,1) 4, if (Xn, Xn-1) = (2, 2). (a) Demonstrate that {Yn: n> 1} satisfies the Markov condition. (b) Evaluate the one-step transition probability matrix for {Yn: n > 1}. (c) Calculate the probability of the weather being sunny in 2 days' time, given that it is sunny today and was rainy yesterday. (d) Find the stationary distribution of {Yn: n >1}. (e) Show that your answer to (d) is a limiting distribution for {Ynin > 1}. (f) Using your answer to (e), find the limiting distribution for {Xn: n >0}, and interpret this