Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,022 questions
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Probability calculation, system uptime, likelihood of occurence.
A little stumped! This is probably a very basic probability question, but I am lost.
At work I was asked the probability of a user hitting an outage on the website. I have some following metrics. ...
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1
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Self Avoiding Walk Enumerations
Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic ...
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Can Gauss sums derandomize any heuristic arguments?
I've always been fascinated by the fact that the classical Gauss sum has absolute value $\sqrt p$, which is exactly what we would expect if we were to interpret the Gauss sum as a random walk. In ...
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Probability theory over noncommutative ring? [closed]
Observation: Entropy is a metric over some non commutative ring. Indeed, if we exponentiate the standard entropy definition
$\displaystyle H(X) = -\sum_{x \in \mathcal{X}} p(x) \ln p(x).$
we'll get
...
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maximum variance unfolding
Consider positive weights $\pi_1, \ldots, \pi_n$ (one can suppose that they add up to $1$) and $n-1$ lengths $d_1, \ldots, d_{n-1}$.
Is there an analytical solution to the following problem:
find the ...
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1
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Approximating expectation [closed]
if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points:
E[f]=(1/N)(summation of f(x) over these N points).
...
3
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Laplace transform of a stopping time for stochastic volatility models
Let $V_t$ be a solution of the SDE
$$dV_t=V_t(rdt+\sigma_t dW_t) $$
where $\sigma_t$ satisfies some other SDE
$$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\\ \prime}_t $$
and $W_t$ and $...
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3
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Expected second moment for random points on a circle
Let $S$ be a circle with unit circumference. Suppose that $n$ random points are chosen independently uniformly from $S$; choosing one arbitrarily as $x_1$, label the rest $x_2, \dots, x_n$ in ...
- 12.8k
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Time integrals of diffusion processes
I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes.
Suppose $X$ is an Ito diffusion process with dynamics
$dX_t = \mu(X_t)dt + \sigma(X_t)...
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Perimeters of random-walk polygons
I have a random walk on $\mathbb{Z}^2$ that takes a step
with equal probability in the three directions that avoid
retracing the previous step.
The walk proceeds until it returns to a lattice point
...
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Does generator of continuous time random walk map heat kernel from L^2 to L^2?
Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi_{xy}$, where, given $e=\{x,y\}\in E$, we write $\...
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Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
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Liouville property in Z^d [closed]
It is well known that $\mathbb{Z}^d$ has Liouville property, i. e. every bounded harmonic function on this graph is constant.
(harmonic means that the value of $f$ in a point $x$ is equal to the ...
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Non-diagonalizable doubly stochastic matrices
Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
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Distribution of the spectrum of large non-negative matrices
This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.
When considering (entrywise) ...
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Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...
- 25.1k
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Exist closed forms of the distribution of return time in markov chains?
Hi, I am interested in the distribution of return times in simple random walks on finite graphs.
Let $G$ be a connected finite graph with, with two independent random walks. If both random walks ...
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answer
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Has the technique of "sprinkling" been used in studying random matrices?
In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...
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"X \in \cdot" in Probability Measure [closed]
My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in:
$$\lim_{n\to\inf}||P(S_n\in\cdot)-P(S_n+k\in\cdot)...
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A point process for modeling location of trees in an infinite forest?
I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process ...
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There is mathematics behind the 1989 Tour de France !
The $1989$ Tour was won by Greg Lemond (USA, $1961$ - ), who beat Laurent Fignon (France, $1960$ - $2010$) by $8''$. Yes, eight seconds! The closest tour in history.
Let me recall a few rules ...
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For what sub-$\sigma$-algebra are these two measures equivalent?
In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
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What is the probability that the range of a set of N randomly chosen real numbers in [0, 1] is less than the reciprocal of N?
(Random number with uniform distribution over [0, 1])
For clarification, in the case where N = 2, we can use geometric probability. On the coordinate plane consider points with 0<=x,y<=1. The ...
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Counterexample Markov process
Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have
$$
P_x[X_t\in A] = 0
$$
for all $t\in [0,T]$ but
$$
...
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Change of measure Markov process
We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$.
I would like to find the conditions ...
- 1,655
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Change of time or change of measure
Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find ...
- 1,655
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What is the expected length of the sum of vectors in a multi-dimensional sphere?
Suppose we pick $m$ vectors i.i.d from the surface of a $d$-dimensional unit sphere (they all have length 1).
What would be the expected length of their sum?
Equivalently, we can ask about the ...
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Random parking problem on a probability distribution
Rényi's Parking Constants comes up when one puts down unit length cars on a interval, such that the probability of covering any two interval is the same.
Are there any published results when the ...
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1
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Calculating $E[X^2Y^2]$ given $E[X^2]$, $E[Y^2]$, $E[X]$, $E[Y]$, and that $X$, $Y$ are Gaussian. [closed]
Suppose $E[X]=E[Y]=0$, and $E[X^2]=E[Y^2]=1$. Can you show that $E[X^2Y^2] = 1 + 2\operatorname{cov}(X,Y)^2$? I am not even sure if this expression is correct, I found it in a geostatistics paper, ...
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Expected value as decision criterion in the context of rare events
I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a ...
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2
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744
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Order statistics: probability random variable is k-th out of n when ordered.
Given a random variable $X_1$ drawn from a distribution with cdf $F$, and random variables $X_2, \cdots,X_n$ drawn from another distribution with cdf $G$, what is the formula for the probability that $...
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Why only three classical matrix ensembles in random matrix theory?
I am just starting out on understanding random matrix theory from a background in applied mathematics. I have a very basic question about the Gaussian ensembles: why are there only three classical ...
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Average wait time for multiple queues where arrivals enter shortest queue
I have been able to find, and understand reasonably well, expressions and derivations for the average wait times for (1) $s$ independent $M/M/1$ queues each with arrival rate $\lambda/s$ and service ...
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2
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Optimally directing switches for a random walk
If you are sometimes called upon directing a random walk in a directed graph, how should you direct it so as to maximize the probability it goes where you want?
Formal statement
More specifically, ...
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2
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570
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How to reading of an integral? Bernoulli trials with variable success rate, p
I have a Bernoulli trial with success rate $p$ and failure rate $1-p$ the odds of $k$ successes is $\binom{N}{k} p^k (1-p)^{N-k}$. I need to evaluate an integral
$$ \int_0^1 dp p^k (1-p)^{N-k} = \...
- 22.8k
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1
answer
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Generalizations of a product formula for the gamma function
Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
\...
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2
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Is stopped brownian motion not a martingale?
In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows.
(...
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Question about Banach's matchbox problem.
Hi,
I've been struggling with this for awhile ( http://en.wikipedia.org/wiki/Banach%27s_matchbox_problem)
and I put together this little bit of Python code
...
- 123
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votes
1
answer
640
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Reachability for Markov process
Let $X$ be a Markov process (in continuous or discrete time) and define an event
$$
R(T,A) = (\exists t\leq T: X_t \in A).
$$
I have seen in one paper that
$$
\Pr[R(\infty,A)] = \sup\limits_{\tau} \...
- 1,655
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655
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Random walks on graphs: Cover time and blanket time
Winkler and Zuckerman conjectured that the blanket time is within a constant factor of the cover time. The conjecture was recently proved. The cover time $C$ is the expectation of the first time $t$ ...
- 121
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3
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A type of stochastic jump process
Let $X \geq 1$ be a integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we ...
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6
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Secret Santa (expected no of cycles in a random permutation)
In a Secret Santa game, each of $n$ players puts their name into a hat and then each player picks a name from the hat, who they buy a Christmas present for. Obviously, if someone picks their own name ...
- 823
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votes
3
answers
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Statistics of a simple Markov chain
Imagine a two-state Markov chain which hops between the states $\pm 1$ with probability $p<1/2$, so that the autocorrelation function after $k$ steps is
$\rho_k = (2p-1)^k$
If I take an ...
- 823
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1
answer
360
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A random variable in a game of knights and queens
Suppose that a game is played on an $n \times n$ board as follows. There are two players, Player 1 has (only) $Q$ queens and Player 2 has only $K$ knights. Suppose that $Q, K \leq n/3$. The game is ...
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Probability theory and measuring the true strength of chessplayers
If you wanted to measure the strength of, say, a chess player, the best way would involve knowing the true value of each position: then you could compute the frequency $W$ with which the player finds ...
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4
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2
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744
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Properties of a continuous-time semi-Markov process as t -> \infty
I am interested in calculating properties of a continuous-time random walk problem which I believe is a type of semi-Markov process.
I have states of the form $n_\pm \in \mathbb{Z} \times \{ +, -\}$. ...
- 143
7
votes
3
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475
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comparing diffusions
Consider a probability distribution $\pi$ on the real axis that has a density (w.r.t Lebesgue) proportional to $e^{-V(x)}$, where $V(\cdot)$ is a potential function. For any reasonable volatility ...
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4
votes
0
answers
580
views
Monotonic properties of harmonic functions on graphs
I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...
- 965
11
votes
3
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743
views
Rainbow matchings (in random graphs)
Suppose we have an $(n,n)$-bipartite graph with edges colored with $k$ colors. Is anything known about the existence of rainbow matchings (i.e. a matching that uses each color exactly once, for $k=n$) ...
- 3,323
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votes
4
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823
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Sweep-segment bot: Will this random walk sweep the plane?
This model is inspired by the random behavior of the
Roomba sweeping robot.
Let a unit segment $ab$ in the plane be placed
initially with $a=(0,0)$ and $b=(1,0)$.
The segment is first rotated a ...
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