# Tagged Questions

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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### 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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### 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 = X_{n-1}...
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### 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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### 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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### 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 time ...
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### 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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### 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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### 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 $x\in \mathbb{X}$. True ...
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### 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 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 $X_n$ ($\mathbb{I}_{A}$...
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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 time ...
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### 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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### 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 http://www.ucl.ac.uk/...
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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 work ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 \\ ...
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### If $A$ is the generator of $(P_t)$, then $A+f$ is the generator of $(P_t^f)$

Let $X=(X_t)_{t\geq0}$ be a Markov process on a state space $\Gamma$ (a Hausdorff topological vector space), let $A$ be the infinitesimal generator of $X$ and let $\mathcal C(\Gamma)$ the space of ...
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### How to find the stationary distribution of a discrete time continuous state space Markov chain

How do I find the stationary distribution of a discrete time continuous state space Markov chain? If the state space is $S$, and if the conditional probability density function of state $y\in S$ ...
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### Busy times for a M/M/1 queue

I have an M/M/1 queue with people arriving Poisson with parameter $\lambda$ and a service time exponentially distributed rate $\mu$. I have been asked to find the average time between the first ...
I'm studying a proof of a large deviations principle and I'm having trouble with a part that is concerned with the semigroup property of a family of operators. Assumptions $(X_t)_{t\geq0}$ is a ...
Consider a continuous-time Markov chain (CTMC) $X$ on a countably infinite state space $S$. The CTMC is irreducible and all the states are positive recurrent. Let $T(i,j)$ be the first passage time to ...