In this exercise, we will build and analyze a simple model for predicting weather using MATLAB. Assume that in Vancouver the weather can be characterized as either sunny, rainy, snowy, or stormy. Suppose analysis of historical records indicates that: 1. if it is sunny today then 35% chance of sunny tomorrow; 55% chance of rainy tomorrow; 10% chance of snowy tomorrow; 2. if it is rainy today then 10% chance of sunny tomorrow; 60% chance of rainy tomorrow; 25% chance of snowy tomorrow; 3. if it is snowy today then 25% chance of sunny tomorrow; 50% chance of rainy tomorrow; 15% chance of snowy tomorrow; 4. if it is stormy today then 40% chance of sunny tomorrow; 30% chance of rainy tomorrow; 0% chance of snowy tomorrow; Let us label the four states sun, rain, snow and storm as 1, 2, 3, 4 respectively. Now define the vector xn = ? ??? xn,1 xn,2 xn,3 xn,4 ? ??? as the state of the system on day n in the future. The entries of xn represent the probability that the system is in any one of its states on day n, e.g. the probability that it is raining on day n is xn,2. If today is day 0, and we know that it is rainy today, then x0,2 = 1 .

This system can be modeled by our random walk framework. The transition matrix for this random walk with the given ordering of states is P = ? ??? 0.35 0.1 0.25 0.4 0.55 0.6 0.50 0.3 0.10 0.25 0.15 0.0 ??? ??? ??? ??? ? ??? (a) Use the information you have been given to figure out the missing row in the above matrix. Report the last row. Explain the reason of your result. (b) Suppose today is a sunny day. Show how you represent x0. What is the probability that it will be snowing 2 days from now, 4 days from now and 6 days from now? (c) Suppose I want to predict the weather after a very long time. Perhaps next year. Perhaps in ten years time. Consider P 300 -the transition matrix that jumps us three hundred days into the future. What do you notice about the matrix besides its columns sum to 1? What does your conclusion imply? If you have difficulty with this last question, try thinking about the probability of different weather in 300 days, starting from sunny, rainy, snowy, or stormy today.