Can someone explain the process of getting the transition matrix bellow? Thanks!
ID: 3331003 • Letter: C
Question
Can someone explain the process of getting the transition matrix bellow? Thanks!
Suppose that whether depends on previous weather conditions through the last two days. Example 4.4 (Transforming a Process into a Markov Chain) Specifically, suppose that if it has rained for the past two days, then it will rain tomor- row with probability 0.7; if it rained today but not yesterday, then it will rain tomorrow with probability 0.5; if it rained yesterday but not today, then it will rain tomorrow with probability 0.4; if it has not rained in the past two days, then it will rain tomorrow with probability 0.2. If we let the state at time n depend only on whether or not it is raining at time n, then the preceding model is not a Markov chain (why not?). However, we can transform this model into a Markov chain by saying that the state at any time is determined by the weather conditions during both that day and the previous day. In other words, we can say that the process is in state 0 state 1 state 2 state 3 if it rained both today and yesterday, if it rained today but not yesterday, if it rained yesterday but not today, if it did not rain either yesterday or today. The preceding would then represent a four-state Markov chain having a transition probability matrix 0.7 0 0.3 0 0.5 0 0.5 0 0 0.4 0 0.6 0 0.2 0 0.8 You should carefully check the matrix P, and make sure you understand how it was obtainedExplanation / Answer
Pij represent probability of state i to state j
now here
state = 0 ,1,2,3
0 - rained both yesterday and today
1 - rained today but not yesterday
2- rained yesterday but not today
3- did not rain today or yesterday
note we are given 4 probabilities 0.7,0.5 ,0.4,0.2
look at first two columns
it is given that 0.7 is probability that it will rain tomorrow if it has rained for past two days (today and yesterday)
now earlier state is 0
leter state is 0 again , as it will be rained tomorrow as well as today {i am writing this with respect to present day ) (consecutive two days)
hence P11 = 0.7
P12 = 0 , because if it has rained today , then it is not possible that with respect to tommorow it will not rain yesterday (that is today)
P14 = 0 similar reason as P12 = 0
P13 = 1 - (p11+p12+P14) = 1-0.7 = 0.3 {sum of any row = 1}
It is similar for other rows .
note that first two columns , we have used 0.7,0.5, , 0.4 and 0.2
and using this and noting there will always be 2 0's in each row , and sum =1
we get required transistion matrix
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