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Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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1answer
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Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes. Let $\mathrm{Li}(x)$ be the offset logarithmic ...
29
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1answer
3k views

When should we expect Tracy-Widom?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...
27
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1answer
670 views

$\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?

Let $X,Y$ be i.i.d. random variables, $\mathbb{E}[X^4]=1$, what's the best upper bound for $\mathbb{E}[(X-Y)^4]$ ? A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 (X^4+Y^4)$ then take ...
22
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6answers
962 views

Maximizing the expectation of a polynomial function of iid random variables

Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$. Question 1. What is ...
22
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3answers
1k views

On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$? ...
18
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0answers
616 views

Erdos-Kac for squarefree numbers

In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then $$\frac{|\{n \le x : \...
18
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0answers
867 views

On random Dirichlet distributions

Fix a dimension $d\ge2$. Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$. For ...
17
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1answer
942 views

Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

https://math.stackexchange.com/questions/64566/riemanns-zeta-function-and-the-uniform-distribution-on-1-0 Stackexchange isn't getting really excited about this, so here it is. The $n$th cumulant of ...
16
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5answers
2k views

A Normal Distribution Inequality

Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
15
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2answers
767 views

Sum of independent random variables

We know that the sum of two independent normal random variables is again a normal random variable. But is the reverse right? If $X$ and $Y$ are independent random variables satisfying $X+Y$~$N(\mu,\...
15
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2answers
829 views

A probability distribution in n dimensional space which its projection on any line is a uniform distribution?

Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?
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0answers
1k views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$d_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
13
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1answer
17k views

How to compute KL-divergence when PMF contains 0s?

From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula: The way I understand this is to compute the PMFs of two given sample sets ...
13
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2answers
559 views

An inequality for expected value of normally distributed variables

Question. Let $X_1,\dots,X_n$ be random variables with normal distribution. Is it true that $$\mathbb E \prod_{i=1}^nX_i^{2k}\ge\prod_{i=1}^n\mathbb E X_i^{2k}$$for any $k\in\mathbb N$? (The ...
13
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1answer
894 views

Does $P(X_1>X_2)$ and $P(X_1=X_2)$, where $X_1$ and $X_2$ are independent and Poisson distributed, uniquely determine the parameters?

Let $X_1$ and $X_2$ be independent Poisson distributed random variables with parameters $\lambda_1$ and $\lambda_2$, respectively. Let $a = P(X_1 > X_2)$ and $b = P(X_1 = X_2)$. Question: ...
13
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1answer
910 views

Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
12
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1answer
8k views

Euclidian norm of Gaussian vectors

Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert_2$. If $\Sigma = {I}_N$, we recognize ...
12
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2answers
1k views

Should you bet in poker against Darth Vader?

This is a theoretical question about poker-type games. I'm not going to specify the rules. You can consider No Limit Texas Hold'em or some simple theoretical model, where each player holds a number ...
12
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1answer
184 views

Mode of a sum of Bernoulli random variables

Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?
12
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1answer
745 views

2/3 power law in the plane

I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I ...
11
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1answer
1k views

Mean of i.i.d Random Variables With No Expected Value

Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...
11
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2answers
1k views

Concentration bounds for sums of random variables of permutations

I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds. As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
11
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1answer
224 views

Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$. Now, let's fix $\sigma$ and let t vary. Then consider the following expression: $$|\Gamma(\sigma+it)|^2$$ For any choice of $\...
10
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7answers
17k 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 ...
10
votes
2answers
446 views

A Conjecture on the Density of a subset of integers

Let $X$ denote the largest subset of odd integers with the property that every exponent in the prime factorization of any $x \in X$ belongs to $X$. The conjecture states that the density of $X$ among ...
10
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3answers
657 views

Entropy and total variation distance

Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...
10
votes
1answer
763 views

Montgomery's pair correlation function without RH?

In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as $$ F(\alpha) = \frac{1}{N(T)} \sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} \...
10
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2answers
287 views

A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. It is also known that the first moment exists for each of them, ...
10
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2answers
561 views

lower-bound for $Pr[X\geq EX]$

Given n random variables, $X_1, ..., X_n$, each takes value 0 or $a_i \in[0, 1]$. $X = \sum_{i=1}^n X_i$ and $EX \geq 1$ is the expected value of $X$. Can we get a lower-bound for $Pr[X \geq EX]$? It ...
10
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2answers
333 views

Generalizations of the Robbins lemma and Gaussian integration by parts

This is getting no attention, so I'll try this here: The Robbins lemma, named after Herbert Robbins, says that if $X\sim\operatorname{Poisson}(\lambda)$ and $g$ is a function for which $\operatorname{...
10
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1answer
929 views

Talagrand's concentration inequality with limited independence

Is there a version of Talagrand's concentration inequality known when the variables have limited independence. More precisely, Let $F:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $1$-Lipschitz convex ...
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4answers
2k views

What structure is needed to define a Gaussian distribution on a given space?

In most textbooks, the normal distribution is defined on $\mathbb{R}^n$ by specifying its probability density function. This works perfectly well, but it isn't really amenable to generalisation. I'm ...
9
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2answers
1k views

What is the most extreme set 4 or 5 nontransitive n-sided dice?

A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia) For some sets, the ...
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8answers
1k views

Semicircle law universality elsewhere

Wigner's semicircle distribution is: $$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$ Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
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3answers
1k views

Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of $$\mathbb{E}\max_{1\le i\le n}|X_i|$$ and ...
9
votes
2answers
942 views

How to efficiently sample uniformly from the set of p-partitions of an n-set?

Let $n,p \in \mathbb{N}_+$ with $p \leq n.$ Let $\mathcal{P}$ denote the set of partitions of $\{1, \ldots, n\}$ into $p$ nonempty sets. How can I efficiently sample uniformly from $\mathcal{P}$?
9
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2answers
676 views

Random Trigonometric Polynomial

Let $t_{1},t_{2},\ldots, t_{n}$ be i.i.d. real Gaussian random variables of zero mean and variance one. Let $a_{1},a_{2},\ldots, a_{n}$ be positive and fixed real numbers and define the random ...
9
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1answer
576 views

Limit of pushforward measures of random variables is “represented” by a random variable

Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of real random variables $X_n:\Omega\to\mathbb{R}$ such that the pushforward measures $(X_n)_*(\mathbb{P}...
9
votes
1answer
1k views

Generalized central limit theorem

I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X_j)_{j\geqslant 1}$ is a stationary sequence such that $X_i$ ...
9
votes
1answer
153 views

q-versions of the geometric distribution and their names

I'm trying to set straight various $q$-deformations of the standard geometric distribution. The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has $$ \mu_1(X=j)=(1-p)p^j,\...
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2answers
385 views

An extension of Gaussian Isoperimetry

The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
9
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0answers
374 views

Torus Graph Dynamics

Consider the torus graph, or the toroidal grid, which looks like (The graph's vertices are the bold dots). I will discuss only square torus graphs, where there is an equal number of vertices in a "...
8
votes
1answer
741 views

Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
8
votes
3answers
21k views

Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...
8
votes
2answers
844 views

Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that $$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$ ...
8
votes
2answers
858 views

Lower bounds on Kullback-Leibler divergence

This was originally a question on Cross Validated. Are there any (nontrivial) lower bounds on the Kullback-Leibler divergence $KL(f\Vert g)$ between two measures / densities? Informally, I am ...
8
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4answers
818 views

Gaussian distributions as fixed points in Some distribution space

I'm taking a course on topology and probabily. Today, the professor remarked something along the lines of: If you look at the space of probability distributions with $0$ mean and variance $1$, ...
8
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3answers
3k views

Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$

Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but ...
8
votes
2answers
4k views

Convergence of moments implies convergence to normal distribution

I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
8
votes
3answers
569 views

A Variance-Tail Description for Continuous Probability Distributions

Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution. I would like to ask ...