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Problem 6. The damage -1n of a system subjected to wcar is a Markov chain with the transition probability matrix0.7 03 0.6P =The system starts in state 0 and fails ...

Question

Problem 6. The damage -1n of a system subjected to wcar is a Markov chain with the transition probability matrix0.7 03 0.6P =The system starts in state 0 and fails when it first reaches state 2_ Find the mcan time to failure Find the mcan times the system spends in states 0 and 1 bcfore the failure.

Problem 6. The damage -1n of a system subjected to wcar is a Markov chain with the transition probability matrix 0.7 03 0.6 P = The system starts in state 0 and fails when it first reaches state 2_ Find the mcan time to failure Find the mcan times the system spends in states 0 and 1 bcfore the failure.



Answers

Refer back to Exercise 3.
$$
\begin{array}{l}{\text { (a) Construct the one-step transition matrix } \mathbf{P} \text { of this chain. }} \\ {\text { (b) Show that } X_{n}=\text { the machine's state (full, part, broken) on the } n \text { th day is a regular Markov }} \\ {\text { chain. }}\end{array}
$$
$$
\begin{array}{l}{\text { (c) Determine the steady-state probabilities for this chain. }} \\ {\text { (d) On what proportion of days is the machine fully operational? }} \\ {\text { (e) What is the average number of days between breakdowns? }}\end{array}
$$

So, first this problem asked us to check that are matrixes regular stochastic. To do that. We check that it's stochastic. We see all the injuries are between zero and one, so we're good there. And some of the columns are 1.2.8 is one and .6 plus .4 is one. So, we know what stochastic. And to see that it's regular. We see that we have no zero entry. So it must be a regular sarcastic matrix. All right. So the next thing that we want to do is find the steady state vector. Now to do that. We want to take our matrix P. And we want to do I minus P. So, what that matrix is going to be is 0.8, negative 0.8 point negative 0.0.6. It was about six. Let's rewrite that There were a negative .6 And positive .6. And then we want to find the no space of this new matrix. So to find the no space, we're just gonna do Gaussian elimination. All right. So, if we go ahead and start role reducing, let's replace the second row with The first row plus the 2nd row. So we'll give you the first row of the same. So and then when we add the two together we get zeros. So what does this tell us? This tells us that .81 times the first entry of our study state vector Is equal 2.6 times the second entry of our study state vector. Well, if we go ahead and rearrange this to solve for X one, then we take X one is equal to .6 of a .8. Which that means X one Is equal 2.75 times the second entry of our study state back directs too. All right. So, what does that mean? That means that we get a scaled version of our study state vector. All right. That over here. This is gonna be your steady state back there. I'll get a box IT and read or a scaled version of it. We're going to put a one for X2 since that is our dependent variable. So we're gonna put one there And then we're going to put a .75 for Rx one as that is our independent variable. So, we have this scale version of our study state vector. Now to find the actual study state vector, we need to make this vector of probability distribution. So, what I mean by that is all the entrance of the vector must sum to warm. So to do that, we're going to need to scale this factor down. So let's rewrite this vector over here to the left and then to find the scaling factor. That's going to make this vector sum to one. We're going to add all the entries of our vector. So we're going to do one plus 0.75 which is equal To 1.75. or I'm going to write this in terms of force as a fraction, This is equivalently equal to seven force already. So now to write our distribution, we are going to multiply our steady state vector by seven force and I'm going to rewrite our study state vector. Also in terms of fractions, it is going to be a little bit easier to see. We're gonna get three forces are original vector And one. We're going to multiply this all by 7/4. And when we multiply every entry, sorry, we're gonna multiply it by 47 because we want to divide each entry by the scaling factor. So you find the scaling factor and then you're going to multiply by the reciprocal in this case. So we have a 4/7 and that is just going to be equal to our final answer, Which is going to be 3/7 and four sevens. And this is our steady state vector. Now we check it all sums to one, so it is a probability distribution. So um we know that we have a valid steady state vector.

So for part a we want to know what the 6th 7th entry represents. Well, the 6th, 7th entry is here in the bottom right corner, so that tells us it represents the probability Of starting in ST two as we were in the second column and ending in state to since it is in the 2nd row. So that means part A. The 6th, 7th is telling us the probability of starting into and going to to now for part B, we want to know what the zero represents. Well, we know that the zero is in the top left entry. Well, so this means that we are starting in the first day and ending in the first state. So the zero is the probability that we start in ST one And we end in ST one and that's going to be our answer to part B. Now, in part C. Our question is, if we are in ST one, what's the probability that we're going to be in ST one and the next spot? What to do that? We look okay. The State one. Call them and we want to see what the probability entry when we are going to state one on the next steps. So is this entry again in the top left corner? Well, from part B, we knew that was zero. So that means for part C. We know that the probability of going from State one to State one is just zero. Now, for our last question, let's say there's only a 50% chance that we start in ST one Then what's the probability that we're going to end up in state 2? The way that we do this is we take the chance that we start in ST 12.5 And multiply it by the probability that we start in ST one. And we ended state too. And then we add that 2.5 times the probability we started in State two and ended in State two And that .5 is because there's a 50% chance we started in state two. Now if we go ahead and plug in these numbers, we see that the probability of going from State one to State two is one, so this is just going to be a 10.5 times one. And then the probability that we start in state two and we go to ST to was 6/7. And we knew that from part A. So now if we go ahead and plug this into a calculator, we end up getting approximately .92, or exactly 13/14. And so this is going to be our answer to party.


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