A stochastic process satisfying the Markov property: the distribution of the future states given the value of the current state does not depend on the past states. Use this tag for general state space processes (both discrete and continuous times); use (markov-chains) for countable state space ...

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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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How to use the Markov property of Brownian motion

This is a problem from Durrett's probability with examples, exercise 8.2.1. It is not homework. The exercise states: Let $T_0 = \inf\{s > 0 : B_s = 0\}$ and let $R = \inf\{t > 1 : B_t = 0\}$. ...
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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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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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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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18 views

Distribution of the first exit time of a one-dimensional diffusiom/ Brownian motion

I have a one-dimensional diffusion on $[0,1]$ and I need to calculate the distribution of the first exit time of the interval $(-\epsilon,\epsilon)$ for an $\epsilon > 0$. A good first step would ...
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33 views

Prove that $\text{lim}_{\Delta t} \rightarrow 0$ of the transition PDF of a std Weiner process is 0

The transition probability density function of the standard Wiener process is: $$ f(x_2,t_2|x_1,t_1) = \frac{1}{\sqrt{2 \pi (t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2(t_2-t_1)^2}} $$ I know that if Markov ...
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36 views

Random walk on the positive integers with reflecting boundary

Consider a Markov chain $X$ on the positive integers where for each $n$: $$n\longrightarrow 1,\;2,\;3\;\dots \;n,\;n+1$$ with equal probability, and $n\longrightarrow m$ with zero probability if ...
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29 views

Constructing transition graph from transition matrix

Ok so for this question I'm having trouble understanding how the transition graph has been drawn from the given transition matrix. This is what I understand and hopefully someone can correct the ...
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33 views

Soft Question - book recommendation - Stochastic Processes

My mother language book on stochastic processes is pretty much complete(~500 pages) but would like another one in English, to have in my library. I'm looking for a similar book containing the ...
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Is it correct to say the expected waiting time is infinite?

Say I have some process that starts in a state in the set of states $S$, each possible starting state having non-zero probability. Some (most) of the starting states eventually result in reaching a ...
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62 views

Prove that $Y_n=X_{n-1}X_n$ is a markov chain

Let $\{X_n\}_{n=0}^\infty$ a sequence of discrete random variables independent identically distributed. Let $Y_n$ such that $Y_n=X_{n-1}X_n$ for all $n\ge 1$ Is $\{Y_n\}_{n=0}^\infty$ a markov ...
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23 views

What is the (practical) difference between a stationary distribution and an equilibrium distribution of a MC?

I know that, for a Markov Chain, a stationary distribution is the (row) vector $\pi$ such that $\pi \cdot P = \pi$, where $P$ is the one-step transition matrix for the MC. Intuitively, I assume that ...
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66 views

prove homogeneous markov chain

$Y_0, Y_1,Y_2,\dots$ are independent and identically distributed random non-negative integer outcomes. Let X_0 = Y_0 $ Let $X_0 = Y_0$ and $X_n = X_{n-1} - Y_n$ if $X_{n-1}>0$, else $X_n = ...
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22 views

How to interpret 'arbitrary customer'

My question is about how to interpret 'arbitrary customer' in the following scenario (see question 2. listed below): "At a single server service station two types of arrivals occur. According to a ...
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30 views

Defective Markov transition matrix and relation to its limiting distributions

Im trying to come to grips with what the physical interpretation of a non diagonalisable Markov Matrix means in terms of what we can deduce about it having a limiting distribution/ what potential ...
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41 views

Theory of Queueing

There's in a banc two identical queues and totally separated : these are two queues of type $M/M/1$. For each of them, the arrivals are separated by exponential times of parameter $\nu$, the ...
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64 views

Renewal process problem, where $X_i$'s are i.i.d. with exponential distribution.

A room is lit by $2$ bulbs. Bulbs are replaced only when both bulbs burn out. Lifetimes of bulb's are i.i.d exponentially distributed with parameter $λ=1$. What fraction of the time is the room only ...
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Prove that $i$ is an accesible state in a markov chain

Let $i$ be a recurrent state of an homogeneous markov chain such that the state $j$ is accesible from $i$ (that is $\exists$ $k\ge 1$ such that $p_{ij}(k)>0$) Prove that $i$ is accesible from $j$ ...
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42 views

Computing hitting times from the stationary distribution

My question is whether it is possible to compute hitting times from the previously calculated stationary distribution $\pi$ of a continuous-time Markov process $(X_t)_{t \geq 0}$. I know that, from ...
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38 views

Mean number of tosses of a fair dice to get a sum of outcomes being a multiple of $5$

Let $S_n$ denote the sum of the outcomes of the $n$ tosses of a fair dice. Let $T=\inf\{n>0: S_n$ is a multiple of $5\}$. Compute $E(T)$ (by means of markov chains). Attempt. Instead of ...
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Mean number of steps needed to reach a recurrent state in a finite irreducible Markov chain

Let $\mathbb{X}$ be a finite state space of an irreducible markov chain $\{X_n\}$ and let $T_x=\inf\{k\geq 0\mid X_k=x\}$ be the number of steps until $\{X_n\}$ reaches state $x\in \mathbb{X}$. ...
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40 views

Markov chains and queues

I do not understand how may I use the Markov Chain $Y$ and and describe the system $X$ using the states that the exercise suggest. I was searching queue's examples and -i understand this is a M/M/1 ...
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Markov process and conditional probability

Given a Markov Process, that is $$\mathbb{P}(X_{n+1}=x_{n+1}\mid X_{n}=x_{n},X_{n-1}=x_{n-1}, \dots, X_0=x_0)=\mathbb{P}(X_{n+1}=x_{n+1}\mid X_{n}=x_{n})$$I Need to prove that ...
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Binary Hidden Markov Model

Consider a binary HMM with 2 observed variables $O_n \in \{0,1\} \; \forall n \in \mathbb{N}$. Suppose that the hidden Markov process $X_n$ is characterised by a known transition probability matrix ...
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Are smooth functions with compact support a core for a generator of a Feller process?

I was reading an article, when the following was stated: Assume we have partial generator defined on some open subset $U\subset \mathbb{R}^n$ $$ A=\sum_{i=1}^n ...
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probability a markov chain leaves a transient state in favor of a recurrent state

Let $\{X_n\}$ be a time-homogenuous markov chain, with state space $\mathbb{X}$ and transition matrix $P$. Let $C_1$ be a transient class and $C_2$ be a recurrent class and let also $x\in ...
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Convergence of mean of an irreducible Markov chain / ergodic theorem

Let $\{X_n\}$ be an irreducible Markov chain on a discrete state space $\mathbb{N}$, that has a stationary distribution $\pi$. Prove or disprove : with probability $1$: $$\lim_{n\rightarrow ...
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55 views

Markov Chain Questions

I've been stuck on these problems for a while. I keep banging my head against the wall, but my calculations are incorrect each time. I sum the probabilities together for each possibility (it's a ...
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Number of $1's$ in a string in terms of successive pairs

Problem. Let $X_n=0$ or $1$ and set $Y_n=(X_n,X_{n+1})$. Set also $\displaystyle \sum_{k=1}^{n}\mathbb{I}_{\{X_k=1\}}$ be the number of times $X_k's$ become $1$, from $X_1$ till $X_n$ ...
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State space extension

In probability theory, it is very common to assume that the sample space is a Borel subset of complete separable metric space, which is basically a Borel space. In probability applications extending ...
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Stationnary distribution - Time of service [closed]

There's in a banc two identical queues and totally separated : these are two queues of type $M/M/1$. For each of them, the arrivals are separated by exponential times of parameter $\nu$, the ...
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24 views

Understanding birth and death process - Stationnary distribution

Customers come to receive a service at a Poisson process of intensity $\lambda$. They are served one at a time and the service time is exponentially distributed parameter $\mu$. In addition, ...
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Branching Process: generation survival or extinction?

Let $p\in [0,1]$, and consider a branching process where the number of offspring of an individual is zero with probability $p$, and is two with probability $1-p$. Initially there is one ...
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Expected infimum of a 1d random walk

Consider a simple symmetric random walk on $\mathbb{Z}$ starting from $0$, $S_n$. Let $I_n := \inf\{S_0, S_1, S_2, \ldots S_n\}$. Is an explicit formula for $E[I_n]$ known?
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Irreducible and recurrent Markov chain - theorem notation question

In [J. R. Norris] Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) (2009), page 35, Theorem 1.7.5 says: In (ii), does it mean $\gamma^k$ is notation for $\gamma^k_i$ ...
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127 views

Stochastic Markov Chain Application: Rat in the maze problem, a modification

I am really new to Stochastic processes, and this is one of the supplementary practice questions that I stumbled across whilst studying: Modify the situation as described in ...
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Markov process - Queuing system - What is the average number of long-term service repairers? [closed]

The operating time of a unit has an exponential law of expectancy $2$ while the repair time follows an exponential law of expectancy $1/2$ . There are $4$ equipments and $2$ repairers who can ...
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26 views

Symmetric Simple Random Walk - Definition Clarification

I'm finding conflicting answers everywhere, including in my own notes. In the phrase "symmetric simple random walk", which part, "symmetric" or "simple" refers to having a probability of $0.5$ to go ...
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Markov DTMC and CTMC: How to build a $Q$ generating matrix using given $2$ state $q$ probabilities? Find Steady State Probabilities

I know that $$q_{(i,j)(i+1,j)}=\lambda_1$$ $$q_{(i,j)(i-1,j+1)}=\lambda_2$$ $$q_{(i,j)(i+1,j-1)}=0.5\lambda_3$$ $$q_{(i,j)(i,j-1)}=0.5\lambda_3$$ where $\lambda_1 , \lambda_2 , \lambda_3 $ are rates ...
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1answer
39 views

How many stationary distributions does a time homogeneous Markov chain have?

I've been given the following definition: For a THMC with one step transition matrix $\mathbf{P}$, the row vector $\mathbf{\pi}$ with elements $(\pi_{i})_{i \in S}$ (where $S$ is the state space) ...
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1answer
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Replacing bulb - Poisson process

A bulb at the entrance of a building has a lifetime of exponential expectancy $100$ days. When burned, the concierge immediately replaced with a new one. In addition, an employee who takes care of the ...
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21 views

Generator - Markov process - Reorganization of the service

Two employees of a brokerage firm receive calls from customers regarding the purchase or sale of mutual funds. When their telephone lines were independent, each was busy a time of exponential ...
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34 views

Create a transition table

I am trying to create a transition table for a markov chain but I have difficulties. Consider a game, where each player (of two, lets call them A and B) has a fixed given probability of scoring 3 ...
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35 views

Sampling uniform equilibrium distribution with Markov Chain Monte Carlo

I'm wanting to sample the discrete uniform distribution over $n = 10$ integers using MCMC. My question concerns the transition probability matrix, $P$. As I understand it, any symmetric, irreducible ...
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Please can someone help me to understand stationary distributions of Markov Chains?

I'm currently trying to understand (intuitively) what a stationary distribution of a Markov Chain is? In our lecture notes, we're given the following definition: This was of little benefit to my ...
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24 views

What does this notation, used in Markov Chains, mean?

In my module on Markov processes, the following notation is used: $$ p_{ij}^{(m,n)} = P(X_n = j \mid X_m = i) \quad \text{where } \: m<n \\ p_j^{(n)} = P(X_n = j) \\ p_{ij}^{(k)} = \: ??? $$ Does ...
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12 views

Generator for the number of shocks to the machine in function - Markov Process

It is assumed that device fails due to a $k$-th shock with probability $\frac{k^2}{9}$ for $k=1, 2, 3$ and, in this case, it is replaced with a new device. If shocks are spaced by independent time ...
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What is a Markovian time evolution model?

Supose I have to constuct a dynamical model for a random variable $X$ . Then I have read that for atmospheric and environmental purposes, a popular and flexible collection of models are (one-step) ...
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An Enquiry Concerning the “reward function” for a Markov Chain

The Statement of the Problem: Consider a Markov chain with state space $$ S = \{1, 2, 3 \} $$ and probability transition matrix $$ P = \left( \begin{matrix} .3 & .7 & 0 \\ ...