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.

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votes
1answer
483 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 ...
0
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2answers
432 views

Chances of streaks in small bit-streams

Let's say a series of 10 bits is output randomly. Now lets do that 256 times. I'd like to find out what the expected number of streaks of 1s or 0s are for each of the possible sizes 1-10. For example,...
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5answers
2k views

Computing correlation between time series with missing data.

Suppose you have two simple Ar[1] series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. ...
4
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2answers
2k views

Formula for the nth convolution of a laplace random variable

Let x_1, x_2, ... be iid draws from a laplace distribution with scale parameter b. Is there a relatively nice closed form for x_1+x_2+...x_n? I've seen a derivation floating around for when b=1, but I ...
12
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4answers
2k views

How long for a simple random walk to exceed $\sqrt{T}$?

Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$. What is an expression for the probability distribution of $...
4
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3answers
536 views

order statistics for components of a random unit vector

Suppose you sample uniformly from the unit vectors in R^n. What are the distributions of the order statistics of the magnitudes of the components of the sampled vectors? That is, for 1 <= i <= n ...
1
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1answer
2k views

Expected value of a gamma-distributed random variable to the n-th power?

Is there a closed form for $E(Y^n)$, where $Y$ is a random variable with a gamma distribution with parameters $\alpha$, $\beta$?
3
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4answers
2k views

Harmonic mean of random variables

The arithmetic mean of normal random variables is normal. The geometric mean of log-normal random variables is log-normal. But is there a common distribution family closed under taking harmonic means?
20
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6answers
2k views

“The” random tree

One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ ...
0
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1answer
363 views

Estimating probability of set membership

I have a number of discrete finite sets, $A_0$ through $A_n$. I do not actually know their contents, but I know the size of each set and the size of the intersection between $A_0$ and each of the ...
147
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33answers
54k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
4
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4answers
879 views

Deconvolution of gamma distributions

If the sum of two independent random variables is gamma distributed does this imply that the individual random variables are also gamma distributed. I suspect that the answer is no, but I do not know ...
6
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7answers
1k views

CLT for stationary sequences with infinte variance

There is a well-known central limit theorem for as a stationary sequences. If $( X_n )_n$ is a sationary sequence and $E X_n=0$ then under suitable mixing conditions the sequence $S_n := n^{-1/2}\...
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2answers
5k views

Examples of random variables

I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
9
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3answers
1k views

When does a pointwise CLT hold?

Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...
8
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2answers
3k views

Laplace transform and fractional moments.

Is there any "easy" way to calculate fractional moments from Laplace transform. To be more specyfic let us consider the following example. Let $X$ be a positive random variable and $L(\theta) := E \...
-2
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2answers
223 views

Evaluate a fair game [closed]

I'm not a mathematician, so my question may be not so clear, sorry. Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 &...
1
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2answers
525 views

Is the min function ever an unbiased estimator for the mean?

Given n iid variables X1 to Xn with an unknown probability distribution, the sample average is an unbiased estimator for the mean of the distribution. Is there some non-trivial probability ...
2
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3answers
578 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 ...
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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
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3answers
995 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
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4answers
723 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
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1answer
455 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]} |...
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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 ...
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1answer
825 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 ...
46
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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
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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
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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
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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
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4answers
952 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
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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
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3answers
688 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-...
64
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12answers
92k 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
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1answer
7k 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
320 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 ≤...
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1answer
422 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
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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 ...
29
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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
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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
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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
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4answers
793 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
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4answers
769 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
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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
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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(...