# Questions tagged [stochastic-processes]

A stochastic process describes the evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples: Gaussian process, Markov process, etc.

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### What does a “closed-form solution” mean?

I have come across the term "closed-form solution" quite often. What does a closed-form solution mean? How does one determine if a close-form solution exists for a given problem? Searching online, I ...
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### Is a time series the same as a stochastic process?

A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?
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### Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
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### Time taken to hit a pattern of heads and tails in a series of coin-tosses

Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
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### What are examples of statistical experiments that allow the calculation of the golden ratio?

There are some very simple experiences that can be done by a kid at home, whose result allows one to statistically approach famous numbers such as $\pi$ or $e$. An example where $\pi$ shows up is ...
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### How likely am I to be descended from a particular person born in the year 1300?

In other words, based on the following, what is p? In order to make this a math problem rather than anthropology or social science, and to simplify the problem, assume that mates are selected with ...
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### How do betting houses determine betting odds for sports?

Let's take football (soccer) for example. There are 3 possible outcomes, home win, draw, away win. I took a random game from bet365 ...
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### What is the difference between Markov chains and Markov processes?

What is the difference between Markov chains and Markov processes? I'm reading conflicting information: sometimes the definition is based on whether the state space is discrete or continuous, and ...
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### What does it mean to say that an event “happens eventually”?

Consider a 1 dimensional random walk on the integers $\mathbb{Z}$ with initial state $x\in\mathbb{Z}$: $$S_n=x+\sum^n_{i=1}\xi_i$$ where the increments $\xi_i$ are I.I.D ...
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### Derivative of a Gaussian Process

I believe that the derivative of a Gaussian process (GP) is a another GP, and so I would like to know if there are closed form equations for the prediction equations of the derivative of a GP? In ...
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### Numeric solvers for stochastic differential equations in R: are there any?

I'm looking for a general, clean and fast (i.e. using C++ routines) R package for simulating paths from a non-homogeneous nonlinear diffusion like (1) using the Euler-Maruyama scheme, the Milstein ...
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### Is stationarity preserved under a linear combination?

Imagine we have two time-series processes, which are stationary, producing: $x_t,y_t$. Is $z_t=\alpha x_t +\beta y_t$, $\forall \alpha, \beta \in \mathbb{R}$ also stationary? Any help would be ...
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### What are the differences between stochastic and fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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### What is the difference between a distribution and a process (Poisson)?

I'm doing my PhD in geomechanics. I thought we use a Poisson-Weibull distribution (for the variability of a parameter at the rock), but reading more about the subject I think maybe is a Poisson-...
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### If a time series is second order stationary, does this imply it is strictly stationary?

A process $X_t$ is strictly stationary if the joint distribution of $X_{t_1},X_{t_2},...,X_{t_m}$is the same as the joint distribution of $X_{t_1+k},X_{t_2+k},...,X_{t_m+k}$ for all $m$, for all $k$ ...
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### The Fishing Problem

Suppose you want to go fishing at the nearby lake from 8AM-8PM. Due to overfishing, a law has been instated that says you may only catch one fish per day. When you catch a fish, you can choose to ...
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### Modifying linear ballistic accumulator (LBA) simulation in R

The "Linear Ballistic Accumulator" model (LBA) is a rather successful model for human behaviour in speeded simple decision tasks. Donkin et al (2009, PDF) provide code that permits estimating the ...
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### How can I show that a random walk is not covariance stationary?

How can I show that a random walk ($y$ follows a random walk) is not covariance stationary? I tried to work on the formula below (with no results) Could you give me just a hint on how to proceed ...
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### Expected number of coin tosses to get N consecutive, given M consecutive

Interviewstreet had their second CodeSprint in January that included the question below. The programmatic answer is posted but doesn't include a statistical explanation. (You can see the original ...
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### Example of a process that is 2nd order stationary but not strictly stationary

Does anybody have a nice example of a stochastic process that is 2nd-order stationary, but is not strictly stationary?
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### How to test if “previous state” has influence on “subsequent state” in R

Imagine a situation: We have historical records (20 years) of three mines. Does the presence of silver increases the probability of finding gold in next year? How to test such question? Here is ...
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### Density of robots doing random walk in an infinite random geometric graph

Consider an infinite random geometric graph in which the node locations follow a Poisson point process with density $\rho$ and edges are placed between the nodes that are closer than $d$. Therefore, ...
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### What's the name for a time series with constant mean?

Consider a random process $\{X_t\}$ for which the mean $\mathbb{E}(X_t)$ exists, and is constant, for all times $t$, i.e. $\mathbb{E}(X_t)=\mathbb{E}(X_{t+\tau})$ for all times $t$ and time shifts (or ...
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