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a. Find the steady-state vector for the Markov chain in Exercise $3 .$b. What is the probability that after many days a specific student is ill? Does it matter if that person is ill today?

(a) $x = \left[ \begin{array} { l } { \frac { 9 } { 4 } } \\ { \frac { 1 } { 10 } } \end{array} \right]$(b) no, it does not matter that person ill today. As we can see aboveprobability for a specific student is ill is $\frac { 1 } { 10 }$ .

Calculus 3

Chapter 4

Vector Spaces

Section 9

Applications to Markov Chains

Vectors

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Hello, Numerator. Okay, we're doing number 13, but it's based on number three. So number three were given this stochastic matrix. Um, as you see described right there. We played around with what happens if we start with 20% L on Monday. What happens on Tuesday? Wednesday for the ills. And what if we start with 100%? Uh, well, on Monday, What happens to the Wells on Tuesday? Wednesday. Okay, but now, question 22 I started question 13. Is this? Let's see, I'm just gonna copy this and some pages to 63 actually. Okay. They're just separate Male, but a little bit. It's his question 13. They want you to find the stochastic. I'm sorry, the City state vector. So what we're gonna do is gonna apply, Okay, just a minute. I'm recording. Okay, Stop. Okay. So regardless of what matrix you start off with, let's say we start with 50% Well, in 50% ill, right? It's a probabilistic myth. Probabilistic factor. So I stand up 200%. Okay, there. Go. Let's get rid of this. Let's see what happens if we apply em several times. That's one way to get the study state factor if it exists. So how about four and in range 1 to 10? That's what we get we're gonna do is just to play em again and again and again so m v 061 m of x one x two M o M times x two x three Well, that's just powers of AM times like zero, isn't it? So I'm gonna say, Let's show m to the n times zero and again again. The question was also doesn't matter what x zero starts at. Do you still get the same result will change up zero see what happens. It shouldn't, right, So I'm gonna say em to the N Times x zero. Let's make some more fancy Was print out the power that we're talking about the X one that we're talking about. So let's put a quote there, comma a comma there In a quote I answer, we get so this question 13 but with the output for a setting up the stochastic matrix, there's output. Be the analyzing the hills and see analyzing the wells. Okay, so we start with 50 50 and eso we get x one is 70. 30 x two is 80 20 and so forth. Now is there a limit? While it looks like it might be going a 0.9 point one, So what's the long term effect in the hills? It was. The question looks like 10%. Let's see if that's true. Let's throw in some bigger exponents. Just make sure, um, how about from 10 to 100? Let's suffering from all outlets, print out everything. Alrighty. So let's see 0.89 point one. But eyes getting really, really, really close to 10.9 point one. I think so. 10% ales in long term. Let's go crazy. How about 100 to 1000? Let's print out every 100 every 100th one. Alrighty. So to the 100th power, we get basically the same thing to the 5/100 power. Same thing with some crazy Randall Verity end there, but especially point line, 90% well and 10% ill in the long term. That doesn't matter if we change zero. Let's see, let's make eggs. Zero. Let's start with 100% well, so we get the same thing and question 13 so that that that started 100% well and zero. Il still seems to be going to 00.9 point once a 0.10% ill in the long term. And how about if we start off with How about we start off with 0%? Well, on 100% ill? Does that matter? Let's see. Question 13. No, I don't think so. Alright, Looks good. So that's your answer. Okay. The raid. Hope that was helpful. Look what your homework. There's question 13 right there. That's it. You wanted to look at the question three. I have a separate video. You can look at that. Here's the up for question 13. Right here. So the final result is 10% ill of long term. Okay. Good luck with, um, you guys hope that was helpful.

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