Markov Chain Problem

juancamilo.dorado@gmail.c · science forum beginner Joined: 16 Sep 2005 Posts: 1

I would like to hear opinion on the solution of the following problem.
I have got my own and one from a professor but I am not quite convinced
about it.

Example 4.3 On any given day Gary is either cheerful (C), so-so (S), or
glum
(G). If he is cheerful today, then he will be C, S, or G tomorrow with
respective
probabilities 0.5, 0.4, 0.1. If he is feeling so-so today, then he will
be C, S, or
G tomorrow with probabilities 0.3, 0.4, 0.3. If he is glum today, then
he will be
C, S, or G tomorrow with probabilities 0.2,0.3,0.5.

Letting X, denote Gary's mood on the nth day, then {Xn >= 0) is a
three-state
Markov chain (state 0 = C, state 1= S, state 2 = G) with transition
probability matrix

0.5 0.4 0.1
0.3 0.4 0.3
0.2 0.3 0.5

1 0. In Example 4.3, Gary is currently in a cheerful mood. What is the
probability
that he is not in a glum mood on any of the following three days?
11. In Example 4.3, Gary was in a glum mood four days ago. Given that
he
hasn't felt cheerful in a week, what is the probability he is feeling
glum today?

Mike · science forum addict Joined: 17 Sep 2005 Posts: 74

This is done by brute force and might very well be incorrect. There must be
a better way. Anyone?

Part (a)
Starting from "cheerful" today and applying transition probabilities
multiplicatively for each of the 8 (of 27) series for the next 3 days not
involving "glum" and adding the 8 independent series yields a probability of
0.585.

Part (b)
Similarly, starting from "glum" four days ago and applying transition
probabilities multiplicatively going backward 3 days (using normalized
transpose of the original transition matrix) and also going forward 3 and 4
days (using original transition matrix) yields a conditional probability of
42.9811% of feeling glum today having not felt cheerful in the past week.
Only series not involving "cheerful" need be considered. The probability of
not feeling cheerful during the last week AND feeling glum today is 1.7992%.
The probability of not feeling cheerful during the last week is 4.1859%.
Thus, the conditional probability is the former divided by the latter.

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