Suppose that there are 3 possible states of the weather in the fall: "sunny", "cloudy" and "rainy". Suppose that if the weather is sunny on one day, then on the following day it will be sunny with probability 0.4, cloudy with probability 0.3 and rainy with probability 0.3. Moreover, if the weather is cloudy on a given day, then it will never be sunny on the following day, but it might be cloudy again with a probability of 0.8 or rainy with a probability of 0.2. Finally, if the weather is rainy on any given day, it will be sunny on the following day with a probability of 0.2, cloudy with a probability of 0.3 and rainy with a probability of 0.5. a) Write the transition matrix A corresponding to the Markov chain described above. b) On any given day, what is the transition matrix B describing the probabilities of being in the states "sunny", "cloudy" and "rainy" two days from now?
Suppose that there are 3 possible states of the weather in the fall: "sunny", "cloudy" and "rainy". Suppose that if the weather is sunny on one day, then on the following day it will be sunny with probability 0.4, cloudy with probability 0.3 and rainy with probability 0.3. Moreover, if the weather is cloudy on a given day, then it will never be sunny on the following day, but it might be cloudy again with a probability of 0.8 or rainy with a probability of 0.2. Finally, if the weather is rainy on any given day, it will be sunny on the following day with a probability of 0.2, cloudy with a probability of 0.3 and rainy with a probability of 0.5. a) Write the transition matrix A corresponding to the Markov chain described above. b) On any given day, what is the transition matrix B describing the probabilities of being in the states "sunny", "cloudy" and "rainy" two days from now?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Transcribed Image Text:Suppose that there are 3 possible states of the weather in the fall: "sunny", "cloudy" and "rainy".
Suppose that if the weather is sunny on one day, then on the following day it will be sunny with
probability 0.4, cloudy with probability 0.3 and rainy with probability 0.3. Moreover, if the weather is
cloudy on a given day, then it will never be sunny on the following day, but it might be cloudy again
with a probability of 0.8 or rainy with a probability of 0.2. Finally, if the weather is rainy on any given
day, it will be sunny on the following day with a probability of 0.2, cloudy with a probability of 0.3
and rainy with a probability of 0.5.
a) Write the transition matrix A corresponding to the Markov chain described above.
b) On any given day, what is the transition matrix B describing the probabilities of being in the
states "sunny", "cloudy" and "rainy" two days from now?
c). Explain the interpretation of a mixed state, that is, a vector with all strictly positive coordinates
that sum to one. In particular, if v is a mixed state, what is the meaning of Av?
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