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.
9,023 questions
17
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Random rings linked into one component?
Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly (...
- 151k
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votes
1
answer
787
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Homotopy of random simplicial complexes
A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with ...
- 249
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2
answers
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A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?
- 761
17
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4
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How many dimensions is it safe to get drunk in?
In Michael Lugo's blog post Variations on the drunken-bird theorem, and real-world sightings he wonders (without coming to a conclusion) what the maximum 'safe' number of dimensions to get drunk in ...
17
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2
answers
953
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Convexity of spectral radius of Markov operators, Random walks on non-amenable groups
Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$.
Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication.
Question
Under which conditions can we show that ...
- 171
17
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1
answer
732
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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, ...
- 118k
17
votes
3
answers
923
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Random permutations from Brownian motion
Let $B(t)$ be a Brownian motion. The ordering of $(0, B(1), ..., B(n-1)) $ is a random permutation in $S_n$. This is not uniform for $n>2$ since the probabilities of the identity permutation $[123.....
- 28k
16
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3
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2k
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Integration of a function over 7-sphere
Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$.
The problem is finding or approximating the ...
- 163
16
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5
answers
2k
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Expected value of determinant of simple infinite random matrix
Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$
I would like to ...
- 355
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5
answers
3k
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Simple random walk on a locally finite graph: when is it recurrent?
I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My ...
- 30.3k
16
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2
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Derivative of a random variable
Hi,
If I have two i.i.d random variables $X,Y$ and a parameter $a$. If I define a new random variable $Z(a)=aX+(1-a)Y$.
Does it makes sense to talk about first, second derivative of the random ...
- 161
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2
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1k
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How often two iid variables are close?
Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$,
$$
\liminf_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$
I can prove the result if they have a ...
- 1,352
16
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2
answers
2k
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On mathematical aspects of the most recent Nobel Prize in economics winners' work
Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's Nobel Memorial Prize in ...
- 622
16
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4
answers
3k
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"Uniform probability" on a set of naturals
It's an obvious and well-known fact that there is no uniform probability measure on a set of natural numbers (i.e. the one that gives the same probability to each singleton).
On a recent probability ...
- 349
16
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6
answers
2k
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Optimal pebble-packing shape
Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes.
Q. ...
- 151k
16
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2
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Probability a polynomial has a root which is a root of unity
Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently.
What is the probability that $P(x)$ has a root which is a root of
unity?
...
- 3,377
16
votes
4
answers
1k
views
Random Diophantine polynomials: Percent solvable?
Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random polynomial:...
- 151k
16
votes
8
answers
4k
views
Brownian bridge interpreted as Brownian motion on the circle
Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle?
The Brownian bridge has some strange connections with the Riemann zeta function (see Williams'...
- 1,666
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votes
3
answers
708
views
An inequality for two independent identically distributed random vectors in a normed space
Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$?
Some background information on ...
- 128k
16
votes
5
answers
3k
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Nice examples/arguments that illustrating the coupling method in probability theory
I want some examples that can best illustrate the idea/power/funny of the coupling argument in probability. I think arguments with coupling makes one think in a more probabulistic way. I have a short ...
16
votes
2
answers
995
views
Probability Problem Involving e
I thought of the following probability problem, which seems to have an answer of 1/e, and wonder if someone has an idea as to how to prove this.
Suppose a man has a bottle of vitamin pills and wishes ...
- 163
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2
answers
1k
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Randomly walking a leashed dog
Let a human $h(t)$ random walk on $\mathbb{Z}^2$ by taking a unit-length step at every
time step $t$. A dog $d(t)$ on a leash of length $\lambda$ follows $h(t)$, also
taking a unit-length step at ...
- 151k
16
votes
4
answers
1k
views
Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
- 2,221
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votes
2
answers
2k
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Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?
Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with.
So, my understanding is that category theory and related fields of higher mathematics ...
- 6,853
16
votes
2
answers
3k
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Number of uniform rvs needed to cross a threshold
Let $N(x)$ be the number of uniform random variables (distributed in $[0,1]$) that one needs to add for the sum to cross $x$ ($x > 0$). The expected value of $N(x)$ can be calculated and it is a ...
- 480
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6
answers
3k
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analog of principle of inclusion-exclusion
When I teach elementary probability to my finite math students, a common error is to mix up the concepts of disjointness and independence. At some point I thought that it might be helpful to some ...
- 2,150
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6
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A normal distribution inequality
Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
- 2,239
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3
answers
2k
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A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k
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votes
5
answers
4k
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Is a fair lottery possible?
I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have ...
- 169
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3
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What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$
Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...
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3
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Expected Degree of a vertex in Delaunay Triangulations
Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that ...
- 3,626
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2
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4k
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Is the space of continuous functions from a compact metric space into a Polish space Polish?
Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ ...
user6096
16
votes
1
answer
1k
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Random polycube shapes
I am wondering if it is hopeless to obtain any firm results
on the following model of a "random polycube shape."
First, a polycube in $\mathbb{R}^3$
is a connected face-to-face gluing of unit cubes.
(...
- 151k
16
votes
3
answers
2k
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Solving a modified birthday problem at a glance
Modified Birthday Problem: a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal?
Three (...
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4
answers
597
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The lattice spanned by $m$ random 0-1 vectors of length $n$
Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
- 24.7k
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votes
3
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791
views
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 ...
- 181
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votes
1
answer
397
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Examples of problems in statistics accessible only using information geometry
I am just curious if there are some examples of problems in statistics that are indeed accessible using information geometry while proofs completely avoiding geometry are unknown. In other words, ...
- 269
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1
answer
346
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Annihilating random walkers
Suppose there are several walkers moving randomly on $\mathbb{Z}^2$,
each taking a $(\pm 1,\pm 1)$ step at each time unit.
Whenever two walkers move to the same point, they
annihilate one another. ...
- 151k
16
votes
1
answer
2k
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Normal approximation of tail probability in binomial distribution
My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
- 497
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2
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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 ...
- 52.3k
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votes
1
answer
276
views
Length of the last edge when visiting points by nearest neighbor order
Take $n$ points uniformly in $[0,1] \times [0,1]$. Then pick uniformly $X_0$ one of these points as your starting point. Then let $X_1$ be the nearest neighbor of $X_0$, let $X_2$ be the nearest ...
16
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2
answers
646
views
How to sample uniformly from singular matrices
I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli ...
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3
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Show there is no positive r.v. $U$ such that $\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge (k+1)/2 }]}{\mathbb{E}[U^k]}, \, \forall k \in \mathbb{N}_0$
Let $U$ be a non-negative random variable such that for all $k \in \mathbb{N}_0$
\begin{align}
\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge \frac{k+1}{2} }]}{\mathbb{E}[U^k]}.
\end{align}
In ...
- 671
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1
answer
929
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A simple stochastic game
An individual, henceforth called the runner starts at the center of an open two dimensional square $\Omega$ of side length $r \geq 2$.
At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
- 6,213
16
votes
0
answers
310
views
Randomized Pascal's triangle: What is the average of all the numbers?
This question was posted on MSE. It received some interesting responses, but no definite answer.
Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
- 3,567
16
votes
1
answer
743
views
Inequalities for marginals of distribution on hyperplane
Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
- 156k
16
votes
0
answers
853
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Self-avoiding random walks that always turn
I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...
- 151k
16
votes
0
answers
2k
views
When does a correlated Brownian motion leave a square?
Let $B=(X,Y)$ be a correlated two-dimensional Brownian motion, that is, the components are standard Brownian motions and the covariance between $X_t$ and $Y_t$ is $t\rho$ for some
constant $\rho \in [-...
- 16.4k
16
votes
0
answers
1k
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Optimal monotone families for the discrete isoperimetric inequality
Background: the discrete isoperimetric inequality
Start with a set $X=\{1,2,...,n\}$ of $n$ elements and the family $2^X$ of all subsets of $X$.
For a real number $p$ between zero and one, we consider ...
- 24.7k
15
votes
4
answers
2k
views
Positivity of certain Fourier transform
Is the Fourier transform of the function
$$ f(\xi) = e^{-t|\xi|^{2m}}$$
positive for $t>0$ and $m \in \mathbb{N}_0$?
- 9,853