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.
9,027 questions
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hitting probability for integrated Ornstein-Uhlenbeck process
Consider an Ornstein-Uhlenbeck position process:
$dV_t=dB_t-\lambda V_tdt$
$dX_t=V_tdt$
where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ .
Let $r>0$ and $S_r$ be the ...
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Enumeration of quadrangulations with a boundary and simple faces.
I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadrangle are distinct.
...
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Volume estimates of rooted embedded tree containing certain subtrees.
Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is ...
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Expectation where linearity does not hold
We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude ...
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If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
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How to solve a generalization of the Coupon Collector's problem
The coupon collector's problem is a problem in probability theory that states the following (from wikipedia):
Suppose that there are $n$ coupons, from which coupons are being collected with ...
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Ising model on a cycle
The Ising model on $\mathbb{Z} / 2d\mathbb{Z}$ gives to the configuration $x=(x_0, \ldots, x_{2d-1}) \in \{-1,+1\}^{2d}$ a probability proportional to $\exp\\big(\beta \sum_i x_ix_{i+1} \\big)$. The ...
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what is the probability that a scissor became the champion?
Here is a question from one of my students:
suppose 8 players are in an elimination match. The players are marked with marked with either R (for rock), P (for paper) or S (for scissors). If two ...
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Derivatives through random variables?
Suppose I have some random variable X with probability distribution P(.;theta). Suppose I have a single sample x from this distribution.
Does it make any sense to ask for the derivative of x with ...
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Product of probability densities of the form x^{-t} exp (-ax)
I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, $y>...
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Kalman Filter...Denoising measurement data to track objects
Hi Everyone,
I am about to implement a Kalman Filter in a software.
I found this very helpful article here:
http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx
The example helps a lot, ...
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104
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Proving that a property holds for random sequences with given marginal distribution by rearrangement
I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
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Maximum entropy priors in infinite dimensional spaces
Is there an extension of maximum entropy probability distributions for function spaces?
For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-...
CommunityBot
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More positive pivotal edges than negative ones at critical bond percolation on Z^2?
Consider critical bond percolation on $\mathbb{Z}^2$ inside a fixed rectangle $(0,0) - (an,n), a \geq 1$ and write $A$ for the event that there is an open crossing in the long (left to right) ...
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Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
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random walk returning probability
Consider a two-dimensional random walk, but this time the probabilities are not 1/4, but some values p_1, p_2, p_3, p_4 with $\sum_{i=1}^4 p_i=1$. For example, from (0,0), it goes to (1,0) with p_1, ...
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Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
CommunityBot
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Upper bound on expectation value of the product of two random variables [closed]
Hello,
I am trying to find an upper bound on the expectation value of the product of two random variables.
So suppose x, y are two non-independent random variables, given that I know the distribution ...
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Total variation distance between a Poisson and a distribution with known mean/variance
Suppose that $\mu$ is the law of a Poisson distribution of mean 1, and that $\nu$ is the law of an unknown distribution on the non-negative integers, though I do know that its mean and variance are ...
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Can one combine (join) probabilities from 2 aspects of a related process?
Consider 2 related aspects of a process for prices in a financial market:
time &
return.
Time
Say I've identified a distribution that reasonably models the occurrence of the lengths of price ...
CommunityBot
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2
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147
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limit of functionals on weak convergent random variables
Suppose real value random variables satisfy
$X_{n} \Rightarrow X$ (convergence in distribution)
as $n\to \infty$ in the same probability space
$(\Omega, \mathcal F, \mathbb P)$.
It is well known that ...
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Distance between probability amplitude functions
Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$:
The Wasserstein distance
For $...
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Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
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135
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expected number of shared 1s between two binary strings from a given set
Let say, I have two binary strings with length N, chosen from a set where there are $2^N-K,(K \ge 0)$ independent strings. What would be the expected number of Ones at the same index from two randomly ...
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Exploiting conditional independence working with covariance matrices
I have a Bayesian network where the number of nodes is potentially large. I've conditioned on some of the nodes (observed data) and I'm trying to draw samples from the distribution remaining nodes (...
CommunityBot
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2
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755
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The Odds 3 (or More) Group Elements Commute
Some time ago I asked about the odds 2 group elements commute. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum
$$ \frac{1}{|G|^3} \sum_{g,h,k} \delta([...
CommunityBot
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902
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Product of densities of a wrapped normal distribution
The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...
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Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
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From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?
I assume the following Lemma is either well known or, more probably, a
Corollary of a much stronger well known Theorem, and I would be grateful for a
reference:
For all $\delta\in (0,1)$ and all $\...
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Probability and events [closed]
Hi everyone
The question is the following:
A certain event may or may not take place. So we say that if we focus on it one time, it has a probability p of being satisfied (0 <= p < 1)
If we ...
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617
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Expected number of components with multiple cycles in a subgraph of a square lattice
Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...
- 1,444
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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons
Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that
assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and
assigns an ...
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examples of random variable X independent to each of A and B, but not to (A,B) [closed]
What are some examples of random variables X, A, B such that X is independent to A, and to B, but not to A and B jointly, i.e., X is not independent to (A,B). In other words, $X \perp A$ and $X \perp ...
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Generalized Ito's formula
Consider classical statement of Ito's formula: Let $X$ be a continuous
semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$
is a continuous semimartingale and
$$F(X_t) = F(X_0) + \...
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Lower bound on Bhattacharya distance between independent Gaussian distributions ?
I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions $p_1\sim\...
- 1,902
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Reference request: a conjecture of Rota on positive functions of a random variable
Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows:
Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2, ...
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Convergence of probability measures on a generating field of a sigma-field
Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mathcal{F}$ be a generating field of $\mathcal{B}$. Assume $\mathcal{F}$ is standard, i.e. it is countable, and any normalized, non-negative, ...
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Uniformly random planar map
Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other ...
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What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
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How to deal with this Chicken-And-Egg problem ?
Let's imagine designing an odds pattern for a game, in which players bet for win or lose.
Suppose the probablity of winning is $p$, thus the probablity of losing is $1-p$.
Now imagine $n_1$ people ...
- 1,902
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What fraction of n x n invertible integer matrices contain at least one unit?
The question is simple:
What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
I'm not sure what the correct measure on $...
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Fixed marginals of joint distribution: status
One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left (...
CommunityBot
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Invariant Measures of Markov Chains under Perturbations
This is a more specific version of a question I asked before without much luck. I believe this should be standard perturbation theory, but looking at Kato's book has not helped. Any references would ...
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Representation theorem for continuous uniformly integrable martingales
For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as:
$$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$
for $t\leq u$. I ...
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Results regarding $E[\min X,Y]$. when $X$ and $Y$ are independent, of given distributions.
Working on fairly unrelated stuff, I needed to prove the following, fairly easy results, and I wonder if anyone can provide references to the literature. Not being a probabilist I wouldn't know where ...
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Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
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Stationarity of an Integral Process
Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with
$$...
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414
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Fourier inversion formula for complex-valued random variables?
The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by
$$
\phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu
$$
(or, so says Wikipedia). How does one recover the ...
5
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2
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472
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Product of random diagonals on the unit circle
Let $P_1, P_2, ..., P_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:
$D=\displaystyle \prod_{1\leq i < j \leq n} \...
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Error bounds for truncating a probability distribution based on the entropy?
Heuristic Background
Consider a set of states labeled $n=1,2,...$ in order of non-increasing probability $p(n)$.
The standard Shannon argument gives meaning to the entropy $S$ of $p$ in terms of the ...
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