Stochastic processes (with either discrete or continuous time dependence) on a discrete (finite or countably infinite) state space in which the distribution of the next state depends only on the current state. For Markov processes on continuous state spaces please use (markov-process) instead.

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Finite-state Markov chain

Suppose $X_n$ is a Markov chain with transition probability matrix $p$ where the set of possible states is $S = \{1,2, \ldots, k\}$. If we are given, $X_1$, $X_2, \ldots, X_{2000}$, can we say about ...
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Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function ...
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Expectation of stopping time on a random walk

Assume $X_1 , X_2 , \cdots$ are i.i.d. with distribution Bernouli$(\frac{1}{2})$, i.e., $P(X_i = 0)=P(X_i=1)=\frac{1}{2}$. Denote $S_0 := 0$, $S_n := \sum\limits_{i=1}^n X_i$, and $\tau_{1000} := ...
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22 views

Building a hidden markov model with an absorbing state.

I'm working on trying to implement a hidden markov model to model the affect of a specific protein that can cut an RNA when the ribosome is translating the RNA slowly. Some brief background: The ...
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144 views

Prove that this transformation of a stochastic matrix (or markov chain) is still a stochastic matrix (or markov chain)

Assume to have an $N \times N$ stochastic matrix $W$, where $\sum_j w_{ij} = 1$ and $w_{ij}$ is a generic element on row $i$ column $j$ of the matrix $W$. Moreover you have the following two $N \times ...
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29 views

Snakes and Ladders and Sample Space

for my Data class project we had to play a board game and do an analysis of it. My group chose rehashed version of Snakes and Ladders. I am almost done the majority of the project, but am stuck on ...
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24 views

A relation between first passage time and occupation time

Let's think about a discrete time Markov chain $X_t$ with only one recurrent state. Let $T$ be the random variable that is the number of steps taken from a given state $i$ to the recurrent state (ie. ...
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24 views

Are queues CTMC?

The $M/M/1$ queue have all the properties of the countable state continuous time markov chain. Is any general queue also a countable state CTMC?
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Underlying sample space in a markov chain

I am studying discrete-time Markov chain and I am confused about the very first example. The example is the Gambler's Ruin: Consider a gambling game in which on any turn you win $\$1$ with ...
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19 views

A problem on random q-colourings of a graph for randomly chosen vertex

Here is an exercise from Olle Haggstrom's "Finite Markov Chains and Algorithmic Applications" from the chapter "Fast Convergence of MCMC Algorithms". The exercise is based on random $q$-colorings of ...
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21 views

Reducing sequential correlations in Metropolis Algorithm

In our last lab, we use MCMC method to simulate a walker walking in the phase space. Using the Metropolis method, a walker at its currect position will sample another point inside a cube (centered at ...
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Pure Birth Question. Find the probability that the population at time $t$ is an odd # given it starts at $0$.

Here is the question. Consider a pure birth process $\{X(t) : t ≥ 0\}$ with birth parameters $\lambda_{2n} = α>0$ and $\lambda_{2n+1} =β>0$ for $n∈N$. Compute $Pr\{X(t) \text{ is odd } \mid ...
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25 views

Finding the stationary distribution of specific homogeneous Markov chain and determining its uniqueness

I am presented with $P =\begin{bmatrix} 0.5 & \alpha & \beta \\ \alpha & \beta & 0.5 \\ \beta & 0.5 & \alpha \end{bmatrix}$ where $\alpha+\beta=0.5$ and $\alpha,\beta \in ...
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15 views

Markovian Model: scheduling jobs to servers

I have the following problem. I tried to look at queuing theory, but it probably fits better as a scheduling problem. I have a set of $C$ servers: each one can perform 1 job. Processes arrive ...
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33 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
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Show that a function of a markov chain is not a markov chain [closed]

How do you show that a function, $Y(n)=g(X(n))$ if some Markov chain $X(n)$ cannot be a Markov chain?
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A few questions related to a Markov chain.

In the picture I have a Markov chain, call it $X(n)$. I have a few questions about this Markov chain. First, is it aperiodic? Second, what is the value of $P[X(1)=1,X(2)=0,X(3)=0,X(4)=1\mid ...
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How to find transition matrix in cascaded FSMs?

I am considering a 4-state system Equivalently, I can use a cascade approach to represent the same system as In cascade approach, If I consider independent condition on input on both ...
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For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
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How to make Markov Chain model from sequence of data in MATLAB?

Markov Chain model considers only 1-step transition probabilities i.e. probability distribution of next state depends only on current state and not on previous state. I have a sequence and from that I ...
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Example of a markov chain that has a distribution that converges to some limit.

Can someone give me an example of a Markov chain that has a distribution that converges to some limit which depends on the initial distribution?
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Markov Chain that isn't Irreducible

What is an example of a Markov chain that isn't irreducible but has a unique distribution, such that its distribution converges to that unique invariant distribution for any initial distribution.
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Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
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25 views

How to test quality of probability estimates?

I have a Markov chain model which produces a probability distribution for absorption in 4 possible absorbing states. I.e. the model estimates the probability distribution for a discrete random ...
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Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
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Exercise about Markov Chain

Let be ($S_{n}$) a Markov chain on $\mathbb{Z}$ such that $P(S_{n+1} = x+1| S_{n} = x)=\alpha_{x}$ and $P(S_{n+1} = x-1| S_{n} = x)=1-\alpha_{x}=\beta_{x}$ for all $x \in \mathbb{Z}$ . Let ...
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26 views

What does it mean for an object to not be following a definition based on some implication

I want to get a deeper understanding of what being an object that doesn't follow a definition means in terms of predicates and logical operators. Suppose the following definition of the closedness ...
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54 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
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30 views

probability of which alarm clock goes off first

I'm learning about continuous time markov chains, in the text I am reading, they are setting up the discussion by talking about a series of alarm clocks that are set: Suppose $T_1,...,T_n$ are ...
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29 views

distance from the origin in a simple random walk on $\mathbb Z^2$

let $S_{n}= \sum_{i=1}^{n}X_i$ be a simple random walk on $\mathbb{Z}$, with $S_0 = 0$. $X_i = 1$ with probability $p$ and $X_i = -1$ with probability $1-p$. It can be shown that ...
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Show that a matrix satisfying certain conditions is non-singular

I have a square matrix $A$ satisfying the following conditions: The elements on the diagonal are negative; All other elements are non-negative; All row sums are less than or equal to $0$; There is ...
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Initial point and initial distribution of the Markov chains

I am reading about Markov chains on a general state space and the ergodicity theory. Some of the ergodic theorems are presented when we consider n-step transition probability conditional on initial ...
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21 views

Markov chain - Expectation $+1$?

Let the transition matrix of a markov chain with states $\{0,1,2\}$ : \begin{equation} A=\begin{bmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{2} & ...
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Period of an irreducible Markov Chain is given by the number of eigenvalues with unit modulus

Suppose $\{X_n\}$ is an irreducible Markov Chain on finite state space $S$. Then, the number of eigenvalues of the transition matrix with unit modulus is precisely equal to the period of the chain. ...
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27 views

Stationary distribution vs invariant distribution of a Markov chain

Lets $p$ be a distribution on a finite sample space with $n$ points. I wish to find a transition matrix that is invariant with respect to $p$, that is $$p^T T = p^T$$. The problem is clearly ...
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37 views

Markov dynamic programming recursion

I'm learning Markov dynamic programming problem and it is said that we must use backward recursion to solve MDP problems. My thought is that since in a Markov process, the only existing dependence is ...
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32 views

Mixing time for lazy random walk on hypercube.

I am studying for a probability exam and am having trouble with the following exercise: Let $X_n$ be a lazy random walk on $\{0,1\}^d$ starting at $(0,\ldots,0)$ (stays put with probability ...
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Stability of a dimer on a square grid after $n$ random steps

On a white square grid there are two black cells. Each step consists of each of the cells 'moving' in one of the four directions with equal probability $p_0=1/4$ (a cell can't stay in the same place). ...
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24 views

When can an embedded Markov chain X for a Markov process Y be reducible?

It's pretty widely documented that a Markov process Y is reducible/irreducible if and only if the embedded Markov chain X is reducible/irreducible. However I'm not sure this works in reverse. I'm ...
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30 views

Modeling the joint distribution of stream statistics

I have a question regarding computing the joint discrete probability distribution of statistics in a number stream. I tried searching all previous posts to look at different forms of this problem ...
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1answer
12 views

emmission probabilities in a hidden markov model with 2 states and an alphabet of 4 characters

I'm reading through a text that is describing how to use use hidden markov models to identify areas of biological sequences that correspond to specific biological features. It starts with a simple ...
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41 views

Expected steps in a Markov chain

I have two jars; initially one is empty while the other holds $n$ red and $n$ blue marbles. Every minute, I do the following procedure: take any pair of marbles of different colors but from the same ...
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Random walk with reflection and skips in linear system

Let's take a case of simple and linear Random walk (0, 1...n) with only one absorbing state n and reflecting state -1, which we can define as: P (move right at state i) = 1/2 and P (moving left at ...
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Poisson: Conditional Probability on Pizza order

I am not sure about my answer. In particular, part b of the following question. Pizza orders arrive according to a Poisson process of rate 20 per hour. Orders are independently for a vegetarian ...
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Getting different results with two methods for a Markov Chain

Given the below Markov transition matrix, calculate $p_{0,1}^9$ \begin{matrix} 0.5 & 0.5 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix} Method ...
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Poisson: $P(N(4)-N(2)=5|N(4)=8)$

Let {N(t), t ≥ 0} be a Poisson process of rate 2. Determine: a) $P(N(4)-N(2)=5|N(4)=8)$ Attempt: The conditional poisson distribution is uniformly distributed between the interval [0,4]. Therefore, ...
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This is a Markov Chain?

Consider two irreducible ergodic Markov chains with the same state space $\{0, 1, . . . , N\}$, with transition matrices $P$ and $Q$ and respective stationary distributions $\pi$ and $\rho$. We ...
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Harris Chain and Stationary Distribution

Consider a Harris chain given by $\{X_n\}$ with the following transition function, $X_{n+1}=\max \{0,X_n-b\} $ with probability $p$ and $X_{n+1}=\max \{0,a-\tau\} $ with probability $1-p$, where $\tau ...
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M/M/1 queue derivation: how to “recursively solve in dependence on $p_0$”

I want to sketch out the derivation of the equations for an M/M/1 queue for a presentation I'm giving. I can understand most of the derivation from Willig but I don't understand this section from p10 ...
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What is the difference between “state-transition-matrix” and a transition matrix?

What's the difference between a state-transition-matrix and a transition matrix (say, for an ergodic Markov Chain) that is typically taught in a basic probability theory course? This is the first ...