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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.

2
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3answers
575 views

When do binomial distributions occur?

A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...
11
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5answers
1k views

What is hidden in Hidden Markov Models? [closed]

Why the word "hidden" present in hidden markov model? What exactly is hidden. Whatever is hidden in HMM isn't it hidden in normal Markov Models?
15
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8answers
2k views

How Does Random Noise Typically Look?

How does random noise in the digital world typically look? Suppose you have a memory of n bits, and suppose that a "random noise" hits the memory in such a way that the probability of each bit being ...
8
votes
3answers
989 views

Singularity of sparse random matrices

The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...
17
votes
4answers
719 views

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 ...
2
votes
1answer
442 views

Suprema of stochastic processes

Let X be a continuous stochastic process. I know that (t>s) P(|X(t) - X(s)|>δ) < |t-s|/δ Is it possible to say anything (e.g. estimate the decay of the tail) about Y=sup_{s \in [0,1]} |...
4
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3answers
2k views

question on sigma-fields

Let X,Y to be mappings from the sample space Ω to R and suppose Y is measurable with respect to σ(X), the smallest σ-field that makes X measurable. Does it follow that there exists ...
1
vote
1answer
814 views

Decoupling lemma for the Lambda(p) problem

I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
14
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4answers
6k views

Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
44
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8answers
15k 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
345 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 ...
19
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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
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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
945 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
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3answers
4k 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-...
63
votes
12answers
90k 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.
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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
421 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
792 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
768 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
17k 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(...