An example where people use linear algebra is in long-term weather trends. For s

Question

An example where people use linear algebra is in long-term weather trends. For simplicity, we will assume only two possible weather conditions, sunny and rainy. Suppose that in West City, if the day is sunny, then the probability that the next day will be sunny is 90%. If it's rainy, the probability that it will be rainy the next day is 50%. Important Note: For what follows, you should convert all probabilities to decimal form. For example, 70% should be written as 0.7. if it's sunny, what's the probability that it will be rainy the next day? If It's rainy, what's the probability it will be sunny the next day? We can represent probabilities of weather conditions with 2-vectors. Given (x y), this will say that the probability that the day is sunny is x and the probability that the day is rainy is y. What does (1 0) mean? What does (0 1) mean? We are going to develop a matrix that tells us, given the current conditions, the probabilities for the next day's conditions. Let W : R^2 rightarrow R^2 be the function that takes the probabilities of the current conditions, given as a 2-vcctor, and outputs the probabilities of the next day's conditions. Find W (1 0) and W(0 1). We know from class that {(0 1), (0 1)}is a basis for R^2.

Explanation / Answer

A. P (tomorrow rainy| today sunny) = 0.1

    P(tomorrow sunny| today rainy) = 0.5

B. (1 0)' (' stands for transpose) means today is sunny (unconditionally) and

     (0 1)' means today is rainy (unconditionally)

C. The matrix W is given by

So W (1 0)' = (p q)'

      W(0 1)' = (1-p 1-q)'

p= P(sunny tomorrow|sunny today) 1-p= P(rainy tomorrow| sunny today) q= P(sunny tomorrow|rainy today) 1-q= P(rainy tomorrow|rainy today)

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