Problem 3: 4-state Markov Chain (Question 4 in Ch 2 of the...

Problem 3: 4-state Markov Chain (Question 4 in Ch 2 of the textbook) The quality of wine obtained from a vineyard varies from year to year depending on a combination of factors, some of which are predictable and some of which are not. From historical data extending over the past 100 years, the year-to-year succession pattern was as follows: Graded Year: Graded Year: Graded Year: Graded Year: Poor Fair Good Excellent Previous Year: 2 6 8 4 Poor Previous Year: 6 12 20 2 Fair Previous Year: 9 20 5 0 Good Previous Year: 3 2 1 0 Excellent Formulate the Markov chain for wine quality (including converting the given information into probabilities).

Transcript

00:01 In this problem, we are basically just given a state space with four states, and we're also given all the single -step transition probabilities from all the states to all of the other states.

00:14 That's all laid out in the question.

00:15 And then in part a, if we let x sub -n denote the state at n at stage n, we're asked to construct the one -step transition matrix, or this markov chain.

00:27 So we are given in the question that the probability of going from state one to state one is 0 .9.

00:40 The probability of going from state one to state two is 0 .07.

00:50 From state one to state three is 0 .02.

00:55 And from state one to state four is 0 .01.

01:01 So it may be easiest to actually answer this question by simply filling in the matrix.

01:07 As we go along.

01:12 So p11 goes into entry 1 -1 in the matrix.

01:19 So we have 0 .9, 0 .07, 0 .02, and 0 .01.

01:30 And it's always good to check that these add up to 1.

01:32 It's 0 .01.

01:34 It's always good to check that these add up to 1, and they do indeed add up to 1.

01:38 Every row should add up to 1, because they are the probabilities of going from the state represented by that row number.

01:47 So in this case state number one, to all of the other possible states.

01:53 So we know that it has to go from one state to one of the other possible states in the state space.

01:58 We're also told that going from state two to one happens with probability of 0 .8, and from going from 2 to 3 with probability 0 .15, and 2 to 4 is 0 .05.

02:14 And we can see that these probabilities already add up to 1, so therefore the probability of going from state 2 to 2 is 0.

02:23 Next we are given the transition probability from state 3 to 2 is 0 .7 and 3 to 4 is also 0 .7, or rather 0 .3.

02:39 So those add up to 1.

02:41 So therefore these transition probabilities must be 0.

02:47 And we're also given the transition probability from state 4 to state 4 is 0 .4.

02:56 Otherwise, you go from state 4 to state 3.

03:00 So that means that must be 0 .6.

03:04 So this is the single step transition matrix for this markov chain...

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