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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.

41
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8answers
14k views

When are probability distributions completely determined by their moments?

If two different probability distributions have identical moments, are they equal? I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero. ...
6
votes
3answers
4k views

Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes

This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly ...
9
votes
3answers
344 views

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 ...
18
votes
9answers
3k views

How can I generate random permutations of [n] with k cycles, where k is much larger than log n?

I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both ...
24
votes
5answers
7k views

Examples where Kolmogorov's zero-one law gives probability 0 or 1 but hard to determine which?

Inspired by this question, I was curious about a comment in this article: In many situations, it can be easy to apply Kolmogorov's zero-one law to show that some event has probability 0 or 1, ...
4
votes
2answers
510 views

maximizing function (stochastic calculus)

S is a price process which follows Geometric Brownian motion with no drift: dS=S*vol*dW, vol=const., W is a Wiener process. Define the following ratio: R=E[Max(f(S)-S(T),0)]/E[f(S)], where S(T) is ...
4
votes
4answers
944 views

Distribution of 1-norm for Gaussian Unitary Ensemble

Suppose I uniformly sample matrices X from the Gaussian Unitary Ensemble (GUE) with variance \sigma^2. Consider the Ky-Fan d norm, i.e. the sum of the singular values, of X. Let's call this Z=||X||...
12
votes
3answers
1k views

Expectation of the product of almost independent Gaussians

Let $X_ i$ be copies of the standard real Gaussian random variable $X$. Let $b$ be the expectation of $\log |X|$. Assume that the correlations $EX_ iX_ j$ are bounded by $\delta_ {|i-j|}$ in absolute ...
6
votes
3answers
687 views

'Focusing' the mass of the Probability Density Function for a Random Walk

Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability ...
10
votes
3answers
3k views

Strong Law of Large Numbers for weakly dependent random variables

Let Xi be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists I such that if |i-i'| ≥ I, then Xi and Xi' are independent), and a finite moment-...
62
votes
12answers
89k views

If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...
4
votes
1answer
6k views

What is the difference between a homogeneous stochastic process and a stationary one?

Hello. I am studying stochastic process. here, I don't know what is difference of "the process is homogeneous" and "the process is stationary" I feel confusing. It seems to similar to me.
-2
votes
1answer
319 views

about Function of Random variables [closed]

Hello, I am studying random variables. Question is this: if rv X & a function g is known, what is the pdf of random variable Y = g(x)? in the textbook answer is explained as follows. P[y ≤ Y ≤...
-1
votes
1answer
420 views

multidimensional multinomial density [closed]

I have data set X = {x_1, x_2, \ldots, x_N}, each x_i is a d-dimensional vector, where scalars are from some finite field (In practice they are categories, represented by integers from 1...C). If ...
13
votes
6answers
3k views

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 ...
28
votes
2answers
7k views

Mean minimum distance for N random points on a one-dimensional line

Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long ...
9
votes
3answers
2k views

erfc lower bound

I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following? $$erfc(x) > \frac{ x \exp(-x^...
7
votes
6answers
2k views

Primes are pseudorandom?

I've been reading the wonderful slides by Terry Tao and thought about this question. Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
10
votes
4answers
789 views

easy(?) probability/diff eq. question

I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
9
votes
4answers
767 views

What m minimizes E(|m-X|^3) for a random variable X?

Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x. A couple weeks ago in a technical ...
35
votes
12answers
16k views

Why is it so cool to square numbers (in terms of finding the standard deviation)?

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do $$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. Why ...
2
votes
2answers
339 views

Limit of sequence involving gamma functions

Let G be the gamma function, and b be a constant in (-2,inf). Let H(n, i) = G(i+1+b) * G(n-i+1+b) / [G(i+1) * G(n-i+1)] for integers n > i > 0. Let S(n) = \sum_{i=1}^{i=n-1} H(n, i). Let x_ n = H(...