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Transitions between sleep stages are described in the article "Multinomial Logistic Estimation ofMarkov-Chain Models for Modeling Sleep Architecture in Primary Insomnia Patients" (J. Pharmacokinet. Pharmacodyn., $2010 : 137-155$ . The following one-step transitionprobabilities for the five stages awake (AW), stage 1 sleep (ST1), stage 2 sleep (ST2), slow-wave sleep (SWS), and rapid-eye movement sleep (REM) were obtained from a graph in thearticle:The time index of the Markov chain corresponds to half-hour intervals (i.e., $n=1$ is 30 min afterthe beginning of the study, $n=2$ is 60 min in, etc..). Initially, all patients in the study were awake.$$\begin{array}{l}{\text { (a) Let } v_{0} \text { denote the probability vector for } X_{0}, \text { the initial state of a patient in the sleep study. }} \\ {\text { Determine } \mathbf{v}_{0 .}} \\ {\text { (b) Without performing any matrix computations, determine the distribution of patients' sleep }} \\ {\text { states } 30 \text { min (one time interval) into the study. }}\end{array}$$(c) Determine the distribution of patients' sleep states 4 h into the study. [Hint: What time index corresponds to the 4 -h mark?

Intro Stats / AP Statistics

Chapter 6

Markov Chains

Section 3

Specifying an Initial Distribution

Probability Topics

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Lectures

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12:10

$ REM $ sleep is the phase…

05:31

Consider the following mod…

05:27

Refer back to Exercise 1 i…

06:56

A Markov chain model for c…

03:41

The article "Markov C…

17:30

The article "Utilizat…

17:13

Correlated bit noise. Let …

05:05

Learning $\ln$ an early ar…

07:01

Consider a Markov chain wi…

04:37

Let $P$ be the one-step tr…

In this exercise, we have five states of sleep that are modeled with the mark of chain, and we are provided with be one step transition matrix for this chain and for part A were asked to determine V sub zero, which is the probability vector for X sub zero. So in the question, we're told that initially all patients in the study were awake. So we know that there's a 100% chance that the initial state is the patient is awake. So that's one for that state. And then the other four states would all be zero. So that's a week, S t one s, t two sws and R E M. Next for B were asked without using any matrix computations to determine the distribution of the patients. Sleep State one interval later. So that's 30 minutes later. According to question, That's one transition later. So when you have a one step transition matrix, the distribution, the probability vector for the state that you will be in after one transition is given by the row corresponding to the state that you're currently in. So if you start off awake one transition later, the probability of vector for your state is given by that role. So this corresponds to the first role in our one step transition matrix. So we can say that B one is equal to 0.9 0.9 0.1 and 00 And then for part C were asked to determine the distribution for the patients Sleep states four hours into the study. So when the question we're told that and equals one corresponds to 30 minutes and equals two corresponds to 60 minutes, so every transition is 30 minutes. So four hours into the study means an equals eight. So what? We're looking for his visa eight, which is equal to P sub zero times the one step transition matrix to the exponents eight And calculating this with software, we get the following, and these probabilities correspond to these states. And so this is your distribution vector. For four hours into the study,

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