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
11
votes
7
answers
29k
views
Resultant probability distribution when taking the cosine of gaussian distributed variable
I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the ...
- 113
4
votes
2
answers
407
views
Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator
Let $\xi_t$ be a zero-mean gaussian process on $[0,1]$ with covariance operator $C$.
I would like to better understand the relation between the covariance operator and the regularity of the ...
- 833
0
votes
1
answer
284
views
The density of x_1^n+x_2^n where x_i are Gaussian
We define a process $\chi_k^n=\sum _{i=1}^k x_i^n$ where x_i are iid gaussian processes.
I try to find the distribution of $\chi_k^n$. If k=1 then we get $f(x^n=y)=\frac1n y^{\frac{1-n}{n}}\exp(-y^{2/...
- 1
4
votes
1
answer
539
views
Buffon's needles revisited
Hi,
I recently came across the famous Buffon's needle problem (http://en.wikipedia.org/wiki/Buffon%27s_needle), and there is no doubt that the problem as well as its answer are elegant.
However, the ...
- 169
4
votes
0
answers
1k
views
Change of Time in Stochastic Integral
Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...
- 1,334
11
votes
3
answers
5k
views
connection between the Gaussian and the Cauchy distribution
I have always been surprised by the fact that the quotient of two independent Gaussian random variables is a Cauchy Random variable - as this is often the case, coincidence in mathematics are not ...
- 2,133
0
votes
3
answers
164
views
Transforming to uniform numbers
Hi
I have a time series of probabilites, vector X
I need to convert the probabilites to uniform numbers.
As I understand it if I put the series into the cdf the output is thus uniform.
The problem ...
- 3
3
votes
1
answer
539
views
Probability of generating the symmetric group
The statement is simple:
What is the probability that a set of n-1 transpositions generates the symmetric group, $S_n$?
The motivation is that I remembered reading that this was an open problem ...
- 543
-3
votes
3
answers
628
views
Roulette probability [closed]
I'm looking for some knowledge on probability, I've scoured the net but I can't really grasp the answer.
I was having a discussion with a co-worker about roulette probability. He says that at any ...
3
votes
1
answer
474
views
Analogues of the Golden-Thompson inequality
Are there any analogues of the Golden-Thompson Inequality for moment generating functions? The Golden-Thompson Inequality asserts the following: If $A$ and $B$ are two $n \times n$ Hermitian matrices, ...
- 31
2
votes
1
answer
2k
views
Density function for a multivariate Bernoulli-like distribution
I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, \ldots, X_k$. Suppose I have observed that $\sum_i X_i = n$. In this case I do not want to treat them as ...
- 129
8
votes
4
answers
2k
views
Motivation of Moment Generating Functions
What is the motivation of defining the moment generating function of a random variable $X$ as: $E[e^{tX}]$? I know that one can obtain the mean, second moment, etc.. after computing it. But what was ...
- 71
4
votes
3
answers
506
views
Continuity in intial state of Brownian Motion
$ B = (B_t, \mathcal{F}_t; t\ge 0 ) $ is a 1-d Brownian family on a
measurable space $(\Omega, \mathcal{F})$ with a family of probability
measures $\{\mathbb{P}^x\}$, i.e. $\mathbb{P}^x(B_0
= x) = 1$, ...
- 1,399
4
votes
1
answer
2k
views
Distribution of running maximum of a local martingale
Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)$ be a given
probability space with usual conditions, on which $W$ is a standard
Brownian motion. For $x \ge 0$, consider
$$X(t) = x + \int_0^t \...
- 1,399
5
votes
1
answer
666
views
Question regarding divergence
Let $E$ be a closed and convex set of distributions on a finite set $A$. Let $P',Q'\notin E$ and let $P^{\star},Q^{\star}$ be their respective estimates in $E$ with respect to the KL-divergence, i.e.,...
- 779
-2
votes
1
answer
890
views
Determine noise distribution [closed]
I'm trying to solve the following least squares problem:
$\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$
where $Ax = b$ and $\tilde{b} = b + w$
Question:
How do I determine which probability ...
- 35
1
vote
0
answers
323
views
Law of the sum of order statistics through MCMC
Hi everyone,
I was wondering is there were some known applications of MCMC methods for simulating the law of the sum ( or alternatively the joint law) of the order statistics of a multinomial (...
- 1,334
3
votes
1
answer
610
views
Looking for a probability distribution
Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size 100 000, all set to 0. We calculate at each round a random number modulo 2 ...
- 133
1
vote
0
answers
299
views
Markov Chain Patterns
Hi
I would like to detect repetitive patterns and deviations from these repetitions. I have historical data and can calculate probabilities for the transitions between my many states. I have ...
- 11
5
votes
2
answers
2k
views
Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
- 1,532
5
votes
4
answers
2k
views
Probability of random permutation having certain cycles
Are there any good references (either books or on-line) on the subject of the distribution of various cycle properties amongst permutations, particularly ones containing exact, closed-forms?
For ...
- 524
3
votes
3
answers
2k
views
Recovering joint distribution from marginals
Suppose we have a Markov Random Field P(X1,...,Xn) on graph G. Suppose we know P(Xi,Xj) for every edge (i,j). Can we recover P(X1,...,Xn)?
If G is a tree, then there's a formula for joint (product of ...
- 2,252
47
votes
7
answers
5k
views
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
- 11.4k
5
votes
3
answers
7k
views
Estimate probability( 0 is in the convex hull of N random points ) ?
Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95
for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ?
The application is nearest-neghbour ...
- 265
2
votes
2
answers
487
views
Cover time of weighted graphs
Consider a connected graph $G$ with non-negative weights on each edge. The sum of edge weights is the same for each vertex, call this sum $W$. A random walk on the graph at vertex $u$ transitions an ...
- 89
3
votes
2
answers
360
views
Convergence of a series of orthonormal gaussian variables
Does anyone have an idea how to prove the following? It is a step in the proof of some theorem in a book about gaussian processes.
Let $f_n$ be an orthonormal sequence of gaussian variables. Consider ...
- 170
3
votes
1
answer
1k
views
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. ...
- 1,338
4
votes
4
answers
4k
views
"Square root" of Beta(a,b) distribution
Under what conditions on a and b is there a distribution $f_{a,b}$ such that the product $XY$ of two independent realizations $X$ and $Y$ from $f_{a,b}$ has a Beta(a,b) distribution?
A standard ...
- 143
4
votes
5
answers
2k
views
Does an "efficient" random number generator exist?
Given some number $n$ and a seed number $s$<$n$, I want a random number generator (RNG) that will go through all integers `<$n$ before coming back to $s$. The resulting random number must be ...
15
votes
3
answers
1k
views
How to shuffle a deck by parts?
This question is mainly a curiosity, but comes from a practical experience (all players of Race for the galaxy, for example, must have ask themselves the question).
Assume I have a deck of cards that ...
- 14.4k
1
vote
0
answers
225
views
What is the limiting distribution of local minima of n mod i, for i up to $\sqrt{n}$, as $n \rightarrow \infty$?
The sequence n mod i
Consider the sequence n mod i for i=1...$\sqrt{n}$. If we draw the sequence as an xy-plot, we get a dense triangle (since n mod i < i). More precisely, the limiting density of ...
- 1,906
1
vote
1
answer
707
views
Independence of conditional hitting distribution and hitting time
Suppose given a d-dimensional Brownian motion $B_t$ starting from the origin and a centered ball with radius 1. Define T as the first hitting time of the sphere (boundary of the ball). How can one ...
- 21
21
votes
3
answers
6k
views
Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.
What is the point of $\pi$-systems and
$\mathcal{D}$ / Dynkin /
$\lambda$-systems?
I am an analyst in the process of consolidating my measure theory knowledge before moving on to harder/newer ...
- 1,771
5
votes
2
answers
1k
views
Inequality involving probability measures [closed]
I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck.
An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
- 779
6
votes
2
answers
545
views
Extension of copulas
Let $(X,Y)$ be a random vector. Suppose that the marginal distribution functions of $X$ and $Y$ are known (say $F_1$ and $F_2$). Then the joint law of $(X,Y)$ is given by the following formula:
$$F_{...
- 931
2
votes
2
answers
571
views
The consequence of overlap sharing for the length-distribution of rods randomly placed on a line
Please imagine that one populates a finite line of unit length, or circle with unit length contour (to avoid edge-effects), with $N$ one-dimensional 'rods' such that their LHS-ends, at positions $(p_1,...
20
votes
2
answers
819
views
A probability question related to extremal combinatorics
$k$ people play the following game: person $i$ independently picks a subset $S_i$ of $\{ 1,2,\ldots,n \}$ according to some distribution $p$ on the $2^n$ subsets; each person uses the same ...
- 976
27
votes
2
answers
812
views
What is the right notion of self-dual (two-dimensional) percolation in R^4?
For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
- 7,857
13
votes
2
answers
3k
views
The probabilistic method - reference to less challenging questions
I am teaching a course in combinatorics and large part of it is dedicated to the probabilistic method especially in the case of graphs. The course is an undergraduate level (almost none of the ...
1
vote
1
answer
438
views
Modular Inequality
I'm trying to find a closed form solution to the following probability given two random values $a$ and $b$:
$P(a \mod{p} < b \mod{p}~|~a \mod{q} > b \mod{q},~p \lt q)$
Ideas?
- 13
1
vote
2
answers
623
views
Assume the standard (better to switch) solution of the Monte Hall problem. Then there's the 3-card Monte problem
Ok, I understand and am convinced by the standard solution of the Monte Hall Problem, i.e. it is better to switch doors after Monte opens one, and improve one's probability of winning from 1/3 to 2/3. ...
- 253
178
votes
8
answers
31k
views
Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
- 4,874
6
votes
1
answer
444
views
When does a matrix define a convolution operator on a hypergroup?
Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...
- 5,425
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
5
votes
1
answer
9k
views
entropy of normal distribution [closed]
What is the entropy of a normal distribution with mean 0 and variance \sigma?
Thanks!
- 63
5
votes
1
answer
1k
views
Intuitive "proof" or explanation of a result in Friedman's urn
Let $g, r, a, b$ be positive integers. In Friedman's urn model we have an urn with $r$ red and $g$ green balls in it. In each step we take one ball out of urn, register its color and return it to the ...
- 349
7
votes
5
answers
8k
views
Two-dimensional random walk
Suppose we have a particle in the plane at the origin $(0,0)$. It moves randomly on the integer lattice $Z^2$ to any of the adjacent vertexs with equal probability $1/4$. What's the probability of ...
- 71
4
votes
1
answer
346
views
approximately linear functions -- more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
- 123
7
votes
1
answer
2k
views
approximately linear functions
i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies
$f(x-y)=f(x)-f(y)+const$
then it is necessarily linear.
are there any general ...
- 123
5
votes
3
answers
501
views
Limit probability
You start with a bag of N recognizable balls. You pick them one by one and replace them until they have all been picked up at least once. So when you stop the ball you pick has not been picked before ...
