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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Self Avoiding Walk Pair Correlation
Let $\gamma(i)$ be a self avoiding walk (SAW) on a 2D lattice $L$ (a square lattice for example) starting at a predefined origin ( $\gamma(0)=(0,0)$ ) and having length $n:=\ell(\gamma)$. Furthermore, ...
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when does inner product with fixed vectors determine joint distribution?
Given a random vector $(X_1,X_2)$. If $aX_1 + bX_2$ is Gaussian for all pairs $a,b$, then $(X_1,X_2)$ is jointly normal. More generally, is the following statement true?
If $aX_1 + bX_2$ has the same ...
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Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
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Likelihood function for sequential random variables
Context
Consider the following sequential data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence.):
$Y_1 = X_1^' \beta + \epsilon_1$
...
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deriving angular central gaussian distribution from a multivariate normal distribution
The angular central Gaussian (ACG) distribution on $(p-1)$-dimensional sphere $\mathbb{S}^{p-1}$ for a symmetric positive definite parameter matrix $\mathbf{A}$ is defined as
$$f(\mathbf{x},\mathbf{A}...
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Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
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Time-integral of a smooth, vector-valued function of a planar Brownian bridge
I'm looking for information on how to compute the distribution of the random vector
$$Z = \int_0^t f(B_s) ds$$
where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in \...
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Follow up question, Ornstein-Uhlenbeck Extension with n mean-reversion values
Hi this Question follows after the answer of Douglas Zare to this post :
So let's be given a positive function $V(x)$ which is smooth enough to have Lispchitzian derivative at all point. Moreover ...
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Radon transform and Log-concavity
This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat f(\omega,t)$...
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Maximum of a set of sums of iid random variables
Consider some probability distribution $D$ over non-negative reals with finite expectation $\mu$. Now for any positive $T$ consider sums of $T$ iid random variables drawn from $D$. A single sum of ...
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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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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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When is the function of a median closer to the median of the function than the mean of the function is to the function of the mean?
Background
notation: RV= random variable, $\mu=$ mean $m=$ median
Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.
If $f(...
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Convergence of sets
Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that
$$
\int\limits_E \phi(x,y)dy=1
$$
for all $x\in E$. Let us consider the non-...
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Square of Binomial Coefficient
Background
I'm modeling Genetic Algorithm(GA) with Markov chains and deriving the expression for the expectation of the first hittig time in the MC with 1 absorbing state and $l-1$ transient states. ...
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Reference request: probability / ergodic theory without measure spaces
In his notes on free probability, Terence Tao describes a general approach to non-commutative probability which prioritizes the algebra of random variables above the sample space; I find this ...
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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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When is a 1-block factor of a non-Markovian process Markov?
Let $Y$ be a discrete stationary stochastic process. Suppose that $Y$ is not $n$-step Markov for any positive integer $n$. Let $Z$ be a 1-block factor of $Y$. For what condition on $Y$ or the ...
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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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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).
...
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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 ...
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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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Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector
Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is ...
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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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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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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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Is there a percolation threshold in the hard discs model?
Take a random configuration of $n$ non-overlapping discs of radius $r$ in the unit square $[0,1]^2$. (You could think of this as taking $n$ points uniform randomly in $[r,1-r]^2$ and then restricting ...
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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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what will be the distribution of ratio of correlated gamma distributed random variables?
If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes ...
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Analog of Chebyshev's inequality for higher moments
I have a positive random variable $X$ with $E[X] = 1$ and a small number $k$ more moments bounded by constants:
$$E[(X-1)^i] = O(1) \forall i \in \{2, ..., k\}.$$
I'd like to bound the average of $n$...
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When does the ratio X/Y of two random variables have a finite moment-generating function?
Let $X$ and $Y$ be two positive random variables with $Y < X$; these may be highly correlated. I would like a reasonable condition on $X$ and $Y$ so that the ratio $X/Y$ has a finite moment-...
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Skellam distribution: Deep connection between Poisson distributions and Bessel function?
The probability mass function for the Skellam distribution for a count difference $k=n_1-n_2$ from two Poisson-distributed variables with means $\mu_1$ and $\mu_2$ is given by:
$$
f(k;\mu_1,\mu_2)= ...
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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:
$$\lim_{n\to\inf}||P(S_n\in\cdot)-P(S_n+k\in\cdot)...
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How to choose $L$ size-$m$ subsets of $\{1,\ldots,n\}$ to maximize expected max overlap with another randomly chosen subset?
GIVEN: Positive integers $n,m,L$ and probabilities $p_1, p_2, \ldots, p_n$.
GOAL: Choose $L$ size-$m$ subsets $S_1, S_2, \ldots, S_L$ of $\{1,2,\ldots,n\}$ to maximize $\displaystyle \mathbb{E}[ \...
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Units in Ornstein-Uhlenbeck(OU) process
Take an OU process characterized by
X(0) = x
dX(t) = - a X(t) dt + b dW(t)
where a,b >0. The parameter a is usually interpreted a dissipative term, and b is a ...
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What is the expected length of the sum of vectors in a multi-dimensional sphere?
Suppose we pick $m$ vectors i.i.d from the surface of a $d$-dimensional unit sphere (they all have length 1).
What would be the expected length of their sum?
Equivalently, we can ask about the ...
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A point process for modeling location of trees in an infinite forest?
I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process ...
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Counterexample Markov process
Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have
$$
P_x[X_t\in A] = 0
$$
for all $t\in [0,T]$ but
$$
...
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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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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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Process for a Gamma distribution with non integer shape parameter
I am sampling the distribution of lifetimes of computers participating in massive volunteer computing initiatives (BOINC projects). While a phenomenological Weibull distribution makes a good ...
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Parametric vs Non-parametric Estimation of Quantiles
Motivation
Suppose that we need to estimate the median from a normal distribution with known variance. One non-parametric approach is to use the sample median as an estimator. However, this does not ...
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Combination of probability distributions with maximum relative entropy
How can we figure out the combination of probability mass functions p and q, such that the relative entropy $D(p||q) = \sum p \log \frac{p}{q}$ is maximum? Of course this is a convex function.
I am ...
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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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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 $...
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$E(X_1 | X_1 + X_2)$, where $X_i$ are (integrable) independent infinitely divisible rv's "of the same type"
The following is inspired by this recent question on math.stackexchange.
Two standard exercises in conditional expectation are to find ${\rm E}(X_1|X_1+X_2)$ where:
1) $X_i$, $i=1,2$, are independent $...
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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} = \...
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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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Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
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Brownian Approximation of Downswings of Walks with Positive Drift
I'm interested in the downswings of discrete walks w(t) whose steps are IID, bounded, and have positive mean. A simple example might have steps which are +1 with probability 2/3, and -1 with ...
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