# Tagged Questions

A stochastic process describes 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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### 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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### Random walk with momentum

Consider an integer random walk starting at 0 with the following conditions: The first step is plus or minus 1, with equal probability. Every future step is: 60% likely to be in the same direction ...
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### Why does the supremum of the Brownian bridge have the Kolmogorovâ€“Smirnov distribution?

The Kolmogorovâ€“Smirnov distribution is known from the Kolmogorovâ€“Smirnov test. However, it is also the distribution of the supremum of the Brownian bridge. Since this is far from obvious (to me), I ...
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### What are some techniques for sampling two correlated random variables?

What are some techniques for sampling two correlated random variables: if their probability distributions are parameterized (e.g., log-normal) if they have non-parametric distributions. The data ...
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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 will studying “stochastic processes” help me as a statistician?

I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University. I asked the lecturer how studying such a course would help ...
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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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### Numeric solvers for stochastic differential equations in R: are they 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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### 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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### 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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### 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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### Generalization of Brownian motion to $\alpha$-stable distributions

Brownian motion is constructed as a limit of the sum i.i.d. Gaussian increments. Can one use a non-Gaussian $\alpha$-stable distribution (e.g. the Cauchy distribution) instead, and still construct a ...
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### Is winning a soccer match independent of previous wins\losses?

$\quad$ I have a friend of mine who is a bit of a gambler ask me this question. He is of poor mathematical background, but has a sense of logic and will probably accept a logical answer in the natural ...
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### Origin of strange formula for equilibrium standard deviation

In the paper M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008, in the Appendix on page 46, how does he get equilibrium standard deviation as following: ...
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### Textbooks on the numerical solution of stochastic differential equations

Is there a canonical textbook or textbooks on the use of numerical methods in solving stochastic differential equations that you would recommend?
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### What are the examples for stochastic processes in Electrical Engineering and Computer Science?

I wanted to find out what kind of different usages of stochastic processes theory in EE & CS are out there. For example, I find these kinds of usages interesting: using stochastic signal as ...
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### The expected value of random variable on tosses of a coin

Came across an interesting problem today. You are given a coin and x money, you double money if you get heads and lose half if tails on any toss. What is the expected value of your money in n tries ...
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### Finding the MLE for a univariate exponential Hawkes process

The univariate exponential Hawkes process is a self-exciting point process with an event arrival rate of: $\lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$ where $t_1,..t_n$ ...
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### Example of a 2nd order stationary, but not strictly stationary process

Does anybody have a nice example of a stochastic process that is 2nd-order stationary, but is not strictly stationary?
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### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
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### Taking random items out of a container with replacement

I take vitamins in the morning, but one of them I only take a half tablet. So, I have an initial container with 100 full tablets, and every morning I take out a random tablet. If it's a full ...
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### Intuitive understanding covariance, cross-covariance, auto-/cross-correliation and power spectrum density

I'm currently studying for my finals in basic statistics for my ECE bachelor. While I think I have the math mostly down, I lack intuitive understanding what the numbers actually mean.(Preamble: I'll ...