All Questions
9,497 questions
8
votes
3
answers
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What is the probability that 4 points determine a hemisphere ?
Given 4 points ( not all on the same plane ), what is the probability that a hemisphere exists that passes through all four of them ?
- 383
2
votes
3
answers
3k
views
Is there any random variable which has unbounded fourth moment? [closed]
In many statements in probability, there is an assumption like bounded fourth moment. So is there any random variable which has unbounded fourth moment?
- 55
18
votes
9
answers
25k
views
Why isn't likelihood a probability density function?
I've been trying to get my head around why a likelihood isn't a probability density function. My understanding says that for an event $X$ and a model parameter $m$:
$P(X\mid m)$ is a probability ...
- 283
9
votes
2
answers
560
views
Integrating a simple exponential over the space of matrices that define a metric
I want to interpret an $n\times n$ matrix $D$ as a set of pairwise distances, and assume that $D$ obeys metric properties. Namely, $D_{ii} = 0$, $D_{ij} \geq 0$, $D_{ij} = D_{ji}$ and $D_{ij} \leq D_{...
- 147
4
votes
1
answer
363
views
Efficiently sampling points from an integer lattice.
Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...
- 41
16
votes
0
answers
1k
views
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
2
votes
2
answers
666
views
Methods for choosing a result from a multiple output node Neural Network
I have a MLP with multiple nodes in its output layer which is predicting membership of classes, one class per output node. I am currently using a "winner takes all" rule for determining which output ...
- 121
0
votes
1
answer
307
views
How to estimate the fraction of graphs with small clique among the graphs with certain edges
Among all $n$-vertex graphs with $M$ edges and constant $k$, how to estimate the fraction of graphs of clique less than $k$? Thanks.
- 3
16
votes
2
answers
3k
views
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
5
votes
3
answers
4k
views
Counting lattice points on an n-simplex
Imagine an n-simplex, the solution set for the expression: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where:
$a_1$ through $a_n$ are positive bounded integers
$x_1$ through $x_n$ are ...
5
votes
2
answers
6k
views
Difference between Beta Process and Dirichlet process
I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf
The problem is that from the definition it follows that every ...
- 233
11
votes
5
answers
4k
views
How can I sample uniformly from a surface?
Given an equation of a parametric surface, is there a general way to sample of points uniformly distributed on that surface?
I'm interested in this problem for purposes of visualisation - rather than ...
- 213
7
votes
2
answers
1k
views
An Expectation of Cohen-Lenstra Measure
The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...
- 22.8k
3
votes
2
answers
1k
views
Probability distribution of the median
Suppose we have $2k + 1$ points $a_1, ..., a_{2k+1}$. Each point is uniformly distributed between 0 and N. What is the distribution of the median (i.e. of the k+1-th point) ?
What happens if $a_1, ......
- 41
1
vote
2
answers
1k
views
Can you explain a step in an expectation maximization algorithm in a Nature article?
I am currently going through the following article: http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html
In this article, how did they arrive at the values in the Estimation step (Figure 1 Step ...
- 19
4
votes
3
answers
4k
views
Range of binomial probability, given a certain number of observations?
Let's say I am given $n$ flips of a coin, $k$ of which are heads.
These are iid flips.
Can I say, with probability $p > 1/2$, that the true probability of heads is in range $[p_1, p_2]$ ?
What is ...
- 287
0
votes
2
answers
144
views
Random values and their probability of reoccuring [closed]
I have a web application that prompts users to answer a question when the computer they are using is not recognized. A user complained today saying she is always prompted for the same question. I ...
- 113
39
votes
9
answers
3k
views
The shortest path in first passage percolation
Update (January 17): The problem has now been solved by Daniel Ahlberg and Christopher Hoffman. (Thanks to Matt Kahle for informing us.)
Consider a square planar grid. (The vertices are pair of ...
- 24.7k
3
votes
2
answers
2k
views
Concentration of measure for gaussian inner products
There exists extensive theory for the concentration of Gaussian measure. Through that, it can be easily proved that the square of the $\ell_2$ norm of a length $n$ zero mean Gaussian vector ${\bf x}$ ...
- 49
8
votes
3
answers
2k
views
randomness in nature [closed]
What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
- 307
8
votes
5
answers
1k
views
linear recurrence relations with random coefficients
Are there such things as recurrence equations with random variable coefficients. For example, $$W_n=W_{n-1}+F\cdot W_{n-1}$$ where $F$ is a random variable. I tried to see if I could make sense of it ...
user577
0
votes
2
answers
630
views
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 ...
- 249
1
vote
1
answer
340
views
for a natural exponential family, A is the cumulant function of h?
Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if
$f(x) = h(x) \exp(\langle \theta, x \rangle - A(\theta))$
defines a family of distributions for $X$, parametrized ...
- 922
99
votes
28
answers
14k
views
Probabilistic proofs of analytic facts
What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
- 1,695
14
votes
3
answers
4k
views
How to generate random points in $\ell_p$ balls?
How do I feasibly generate a random sample from an $n$-dimensional $\ell_p$ ball? Specifically, I'm interested in $p=1$ and large $n$. I'm looking for descriptions analogous to the statement for $p=2$:...
- 667
12
votes
2
answers
812
views
Inequality in Gaussian space -- possibly provable by rearrangement?
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
- 6,694
2
votes
3
answers
571
views
How does the Dirichlet process work?
Hi, i'm looking to get into nonparametric bayesian techniques but I'm having problem understanding what's going on in the definition of the Dirichlet process or how it works. So what does P ~ DP(&...
- 23
79
votes
11
answers
21k
views
How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?
My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...
- 804
7
votes
2
answers
627
views
Probability vertices are adjacent in a polygon
With regard to my original question:
A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?
I suppose that the responses ...
- 73
4
votes
2
answers
2k
views
Probability Question
You have $N$ boxes and $M$ balls. The $M$ balls are randomly distributed into the $N$ boxes. What is the expected number of empty boxes?
I came up with this formula:
$\sum_{i=0}^{N}i\binom{N}{i}\...
- 597
2
votes
2
answers
1k
views
Result of repeated applications of the binomial distribution?
What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?
To clarify, an example.
Suppose that a bunch of people are playing a game with k (to ...
- 2,383
2
votes
2
answers
521
views
A variant of the hypergeometric distribution - in the literature?
I have been working on a problem in combinatorics that makes use of the following discrete distribution.
Let $a_{1}, a_{2},..., a_{N}$ be any binary sequence of of length $N$ with $n$ ones and $m$ ...
- 201
5
votes
3
answers
601
views
Monte Carlo simulations
I was wondering what were the models of statistical physics that are still considered difficult/slow to simulate (exactly, or approximately) with the current technology of Monte Carlo approaches. I ...
- 2,133
4
votes
0
answers
497
views
A Local CLT with large variance
For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a
random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
- 263
8
votes
5
answers
757
views
Coupling of Wiener processes
Is it possible to find a coupling of two Wiener processes $W^0, W^x$ (i.e. two Wiener processes defined on a common probability space). One starting from $0$ and the other from $x$ such that
$W_t^0 -...
- 1,491
8
votes
0
answers
348
views
A formula for moments of the limit distribution of singular values in the proof of the circular law
One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix
$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - zI)^\...
4
votes
5
answers
2k
views
Martingales and Betting Strategies
Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...
10
votes
3
answers
644
views
Models with SLE scaling limit
What discrete processes/models have been proven to have scaling limits to $\text{SLE}(\kappa)$, for various $\kappa$?
I know about loop-erased random walk and uniform spanning trees.
What about ...
- 85.6k
3
votes
3
answers
2k
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
- 47
14
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2
answers
783
views
Are two probability distributions uniquely constrained by the sum of their p-norms?
Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, ...
- 2,649
11
votes
1
answer
1k
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measurable sets not depending on even coordinates
Let $A\subset\{0,1\}^\omega$ be a measurable set (w.r.t. the usual borel sigma algebra) which does not depend on any even coordinate (that is, if $x\in A$ and $x$ and $y$ agree except on a finite ...
- 6,711
18
votes
4
answers
6k
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most general way to generate pairwise independent random variables?
Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them?
I'm wondering because I find it difficult to come up with a lot of examples of ...
- 1,093
1
vote
1
answer
336
views
A probabilistic inequality [closed]
Suppose $x_1,x_2,...,x_6$ are non-negative Independent and identically-distributed random variables, is it true that $P(x_1+x_2+x_3+x_4+x_5+x_6 \lt 3\delta) \lt 2P(x_1 \lt \delta)$ for any $\delta \...
- 35
-1
votes
6
answers
2k
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Chances to win an election
Let's say that tomorrow national president election is held. A poll asks 1100 persons which of the two candidates, A or B, will he or she will vote. 750 say will vote A, and 250 say will vote B. What ...
- 330
7
votes
2
answers
648
views
What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?
The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...
- 5,992
2
votes
1
answer
380
views
Parity, Balls and Boxes
Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m ...
- 263
0
votes
1
answer
189
views
Difference Equations & Possible Limits
The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...
13
votes
11
answers
6k
views
A problem of an infinite number of balls and an urn
I think that the following problem originated in a probability textbook :
You have a countably infinite supply of numbered balls at your disposal. They are all labeled with the natural numbers {1,2,3,...
5
votes
4
answers
1k
views
Intuitive explanation to Probability question [closed]
I have \$3. I flip a coin. If I get heads, I get \$1. If I get tails, I lose \$1. The game stops when I have \$0 or \$7. What is the probability I get \$7?
I solved this by creating a system of ...
- 597
2
votes
1
answer
558
views
Which iid variables give a normal variable, when multiplied?
Hello, I hope you'll find my riddle interesting.
Z = XY
Z ~ N(0, 1)
X, Y are iid random variables (independent, identically distributed). We assume X and Y are symmetric.
What is the distribution of ...
