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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Confusion about notation on continuous pathed stochastic processes

In the definition below, I am unsure on what the $[\omega]$ in $X(t)[\omega]$ is meant to signify. Is it saying that we first observe a stochastic process, create the mapping with the seen values of the random variables, then we denote values of…
math111
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Reaching state N in at most n=4 steps

I was working on a question and I doubt the solution given is the right one. To simplify, consider a state space $S = \{1,2,3,4,5,6\}$ and an initial distribution $a^{(0)}=(1,0,0,0,0,0)$ with transition matrix P. The question asks for the…
Marius
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Definition of Probability mass function of a random process.

A random process $X(t)$ is defined as $X(t)=1, 0\leq t \leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$ I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of…
AspiringMat
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Ultrastrators civilization

I came across this question: In the ultrastrators civilization on planet Anachronista, the population is divided into four strata which, in order of status , are labeled Alpha, Betta, Gamma and Delta. By the traditions of the civilization, no child…
Astatine
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Proving that expectation and integration is interchangeable

I am trying to prove that the expectation and integration are interchangeable in the following part of the proof Link to the proof (I am not allowed to embed images yet , sorry) I tried to apply the Tonelli's theorem but that would require that the…
vortex_sparrow
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Let $W_t$ be an one-dimensional Wiener process? Is $W_t$ bounded almost sure?

With the standard one dimensional Wiener process $W_t$? Is $P(\omega: \exists \mbox{ finite} \lim\limits_{t\to \infty}W_t(\omega))=1$? Could you help me! Thank you in advance!
Kae
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Birth and Death Process

Airline passengers must pass through a security check point consisting of a metal detector and carry on luggage X Ray machine. Suppose that passengers arrive according to a Poisson process with a rate ω = 20 passengers per minute. The Security check…
Astatine
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Sample continuity and continuity at every index value

From Wikipedia Let $(Ω, Σ, P)$ be a probability space, let $T$ be some interval of time, and let $X : T × Ω → S$ be a stochastic process. For simplicity, the rest of this article will take the state space $S$ to be the real line $\mathbb{R}$, but…
Tim
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Does a process satisfying this condition for jumping between states necessarily have exponential waiting time?

This is a property of Poisson process. But I will ask about it for a more general process. For a stochastic process $X$ with continuous time and discrete state space, if $\forall i$ in the state space $$ P(X_{t+h}=i | X_t=i) = 1 - r_{ii}(t) h +…
Tim
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Branching Process

Consider a branching process $X=\{X_n, n=0,1,\dotsc\}$ where $X_n=\sum\nolimits_{i = 1}^{{X}_{n-1}}{Z_i }$ , $X_0=1$, and let $Z_i$ be such that $P[{Z_i=0]}=1/2$, $P[Z_i=1]=1/4$, $P[Z_i=2]=1/4$. How to find the probability of extinction …
QAK
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Is a Random/Stochastic process always dependent on 'time'?

Is a Random/Stochastic process always dependent on 'time', or can it be dependent on any other 'unit'? For instance, suppose, We roll a dice whenever a soccer match takes place in a major league in a country in Europe. Now, a soccer match can…
user366312
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Is a linearly ordered set equally spaced out?

When defining a discrete - time stochastic process, the author of An Introduction to Discrete-Valued Time Series states that such a process is a sequence of random variables $(X_t)_{\mathcal{T}}$ where $\mathcal{T}$ is both a discrete and linearly…
Vykta Wakandigara
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how to find the first k events are registered in poisson process

Problem 1.3. As a complication of the Rutherford–Chadwick–Ellis experiment, suppose that the number of particles that decay in each interval is registered by a counter. We assume that events occur according to a Poisson process with intensity λ > 0,…
appleyuchi
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Veryfing a random variable is a stopping time (filtration indexed by natural numbers)

If we want to check whether $\tau$ is a stopping time with respect to some filtration $\lbrace \mathcal{F}_{n}\rbrace$, where $n\in\mathbb{N}$, we can verify that $\lbrace\tau= n\rbrace\in\mathcal{F}_{n}$ for all $n$ rather than $\lbrace\tau\leq…
czachur
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transient and steady state probabilities

The difference between transient and steady state probabilities is only that, transient probabilities are time dependent and steady state probabilities are not?
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