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2 . (5 marks) What is the probability of transition from state 1 to state 3 in two steps, for the Markov chain with transition matrix T0.2 0.6 0.1 0.5 0.1 0.2 0.3 0...

Question

2 . (5 marks) What is the probability of transition from state 1 to state 3 in two steps, for the Markov chain with transition matrix T0.2 0.6 0.1 0.5 0.1 0.2 0.3 0.3 0.7

2 . (5 marks) What is the probability of transition from state 1 to state 3 in two steps, for the Markov chain with transition matrix T0.2 0.6 0.1 0.5 0.1 0.2 0.3 0.3 0.7



Answers

Consider a Markov process with transition matrix CAN'T COPY THE FIGURE (a) What does the entry 0.2 represent? (b) What does the entry 0.1 represent? (c) If the system is in state 1 initially, what is the probability that it will be in state 2 at the next observation? (d) If the system has a $50 \%$ chance of being in state 1 initially, what is the probability that it will be in state 2 at the next observation?

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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