A machine is either working (state 1) or not workind (state 2). If it is working one day the probability that it will be broken the next day is 0.1. If it is not working one day the probability that it will be working the next day is 0.8. Let Tn be the state of the machine n days from now. Assume the Markov assumption is satisfied so that Tn is a Markov Chain.
a) Give the transition matrix P for T.
If machine is working on day 0
prob working on day 1 = .9
prob broken on day 1 = .1
If machine is broken on day 0
prob working on day 1 = .8
prob broken on day 1 = .2
so
working broken 1 = working broken 0 *
.9 .1
.8 .2
for example if working day 0
(doing second row with / because of font text limitation here)
[W1 B1] = [ 1 0 } [ .9/.8 .1/.2 ]
= [.9 .1]
then
W2 B2 = [.9 .1][ .9/.8 .1/.2 ]
= [.81+.08 .09+.02 ]
= [.89 .11 ]
and
Wn Bn = [1 0] *
| .9 .8 |^n
| .8 .2 |
To construct the transition matrix P for the Markov Chain T, we need to determine the probabilities of transitioning from one state to another.
In this case, we have two states: state 1, which represents the machine working, and state 2, which represents the machine not working.
From the given information, we know the following transition probabilities:
1. If the machine is in state 1 (working), the probability of it transitioning to state 1 (working) the next day is 0.9 (since the probability of it being broken the next day is 0.1).
2. If the machine is in state 1 (working), the probability of it transitioning to state 2 (not working) the next day is 0.1.
3. If the machine is in state 2 (not working), the probability of it transitioning to state 1 (working) the next day is 0.8.
4. If the machine is in state 2 (not working), the probability of it transitioning to state 2 (not working) the next day is 0.2 (since the probability of it being broken the next day is 0.2).
Based on these transition probabilities, we can construct the transition matrix P as follows:
| P11 P12 | | 0.9 0.1 |
P = | | = | |
| P21 P22 | | 0.8 0.2 |
Here, P11 represents the probability of transitioning from state 1 to state 1 (working to working), P12 represents the probability of transitioning from state 1 to state 2 (working to not working), P21 represents the probability of transitioning from state 2 to state 1 (not working to working), and P22 represents the probability of transitioning from state 2 to state 2 (not working to not working).
So, the transition matrix P for T is:
| 0.9 0.1 |
P = | |
| 0.8 0.2 |