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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Simplification on the estimation on error of the ratio of 2 random variables

Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables. Distribution of $Z=\dfrac{X}{Y}$ Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
youpilat13's user avatar
4 votes
2 answers
263 views

Using $\delta$-method to "estimate" undefined moments of a random variable?

I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is. Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...
Aaron Hendrickson's user avatar
5 votes
3 answers
594 views

Convergence speed of a random dyadic rational generator

We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$ two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
Penelope Benenati's user avatar
2 votes
0 answers
48 views

Log-concave probability measure with slowest decay

Let $X$ be a real valued random variable with log-concave distribution $\mu$. For each $x \in {\mathbb R}$, let $$ \phi_\mu(x)=\min\limits_{c\in{\mathbb R}}E[e^{c(X-x)}] $$ be the minimal value of the ...
Bogdan's user avatar
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Is the topology generated by the convergence of finite-dimensional distributions metrizable?

Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
vaoy's user avatar
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6 votes
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146 views

Distribution of iid hypergeometric random variables conditioned on the sum

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$....
Brendan McKay's user avatar
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What is the expected value of the sum of the k (out of a set of n) smallest normal random variables?

Given $n$ independent normally distributed random variables $X_1,X_2,...,X_n \sim N(\mu,\sigma)$. For any $k\leq n$, let $X_{(k)}$ be the k-th order statistics (i.e., the k-th smallest value). What is ...
Ron Banner's user avatar
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623 views

sub-exponential type upper bound on the Poisson probability

I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received. Question: For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
Jane's user avatar
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1 answer
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Rate of variance's decrease for the mean's distribution of infinite variance i.i.d. random variables

Consider a set of i.i.d. (positive) random variables $\{X_i\}_{i=1}^N$. Each variable $X_i$ has a distribution with finite mean but infinite variance. In particular, if $P_{X_i}(x)$ is the P.D.F. of ...
user1172131's user avatar
3 votes
3 answers
249 views

Testing uniformity for continuous probability distributions

Suppose I can sample from a random variable $X$ which is distributed on a compact interval, say, $[0,1]$. Fix a distance measure between distributions, say total variation. Let $\epsilon\in(0,1)$. How ...
Sela Fried's user avatar
3 votes
2 answers
364 views

Distribution of a certain functional of iid $N(0,1)$ random variables

Suppose that $X_1,\ldots,X_n$ are iid $N(0,1)$ random variables. Consider the random variable given by $$ \xi_n =\Bigl|\frac1{\sqrt{n}}\sum_{t=1}^nX_t\Bigr|^2-\frac1n\sum_{t=1}^nX_t^2 =\frac1n\sum_{s\...
Cm7F7Bb's user avatar
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3 votes
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Eigenvalues of Hadamard product of two Wishart-type matrices

Given two independent Gaussian matrices with i.i.d. entries: $A\in\mathbb{R}^{n\times p}$ and $B\in\mathbb{R}^{n\times q}$, where and $A_{i,j},B_{i,j}\sim\mathcal{N}(0,1)$. Assume that $\max(p,q)<n....
M-Brust's user avatar
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Simplify Kantorovich–Rubinstein duality when distributions share a common marginal

Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
joemrt's user avatar
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Lebesgue measure of a circular arc on the surface of the hypersphere [closed]

The following sub-problem came up in one of my research works: Suppose that $U$ is a uniform random variable on the surface of the $d$-dimensional sphere with center at the origin and of radius $r$. ...
Probabilist's user avatar
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89 views

Limit of a linear discrete-time stochastic process with uniform noise

I have posted this in the math and stats sites, but I am not sure where the proper forum for this question is. If it is not here, please go on and delete it. Suppose we have a stochastic linear ...
NoobNoob's user avatar
2 votes
1 answer
863 views

measure of a degenerate Gaussian distribution

I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it in a close form. After starting with a Gaussian random variable and restricting it to a condition, I ...
Skull Soul's user avatar
3 votes
1 answer
200 views

Random planes separating points in $\mathbb{R}^3$

We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
Penelope Benenati's user avatar
1 vote
1 answer
153 views

Is it always possible to determine the distribution of a random variable given all its moments? [closed]

we we're asked about it, and I know that answer is "NO", and I haven't found an good enough answer yet and would appreciate an explanation with examples.
Matan Cohen's user avatar
1 vote
1 answer
106 views

Upper bound on the ratio of Poisson CDFs [closed]

Suppose $X \sim Pois(\lambda)$. I'm interested in an upper bound on the ratio, $$\dfrac{P(X \leq n)}{P(X \leq n-1)}\,,\,\,\text{for $n=1,2,3,...$}$$ Observe that, the ratio is $>1$ & as $n \to \...
SL_MathGuy's user avatar
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1 answer
272 views

Joint distribution of dependent Gaussians and their product

Consider a pair of dependent zero mean unit variance Gaussians, $$X,Y \sim \mathcal{MVN}\left(\vec{0},\begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}\right).$$ Their product $Z:=X\cdot Y$ is ...
Richard Border's user avatar
6 votes
1 answer
231 views

Ordering preference for two zero mean Gaussian outcomes

Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
Pierre's user avatar
  • 171
3 votes
1 answer
143 views

Randomized version of Turán's theorem II

$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then \begin{equation} \om(G)\ge\...
Iosif Pinelis's user avatar
5 votes
1 answer
206 views

Randomized version of Turán's theorem

Turán's theorem says the following. Take any natural $n$ and $r$. Suppose that \begin{equation*} |G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0} \end{equation*} where $|G|$ is the number of edges of ...
Iosif Pinelis's user avatar
1 vote
1 answer
188 views

Continuity of pushforward operation

Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.: $$ \sup_{x \in X} d_Y(f(x),g(x))<\epsilon. $$ Then, are their push-forwards close in ...
ABIM's user avatar
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Can I express this random variable in terms of known distributions?

By computing the Laplace transform of the total length of a random tree (the nested Kingman coalescent tree with coalescence rates $\gamma$ for the individuals and $\gamma'$ for the species), we would ...
Josue's user avatar
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1 vote
0 answers
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An Inequality of Expected Value of Random Variables

I encountered the following problem in my research: Suppose there are $N$ random variables that are independent and identically distributed (IID). The probability density function (PDF) of these ...
xiaohaozi's user avatar
0 votes
1 answer
114 views

Perturbative approach starting from a probability distribution approximated form

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$, such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity. Consider the generic ...
user1172131's user avatar
0 votes
0 answers
69 views

Integration of fractional function over Rice distribution

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx \end{equation}...
hichem hb's user avatar
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1 vote
1 answer
116 views

Modulus of continuity of parameterizing Wasserstein

Let $x_1,\dots,x_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w_1,\dots,w_n)\in\Delta$ to the finitely supported ...
Bernard_Karkanidis's user avatar
1 vote
0 answers
201 views

Distribution and expectation of inverse of a random Bernoulli matrix

This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
Nishant Singh's user avatar
2 votes
0 answers
57 views

Maximum entropy distribution in the hyperbolic plane with given "mean" and "variance"

On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy ...
popstack's user avatar
  • 265
5 votes
1 answer
187 views

Probability of gaps between coordinates of a random point on the sphere

Let $X=(X_1,\ldots,X_n)$ be a point chosen uniformly at random from the sphere $S^{n-1}\subseteq \mathbb R^n$. Given $a>0$, what is the probability that $|X_1|^2-|X_i|^2\geq a$ for all $i>1$? Is ...
Hadi's user avatar
  • 731
1 vote
0 answers
43 views

Small parameter expansion of probability density

I am trying to describe the motion of a particle that moves according to the Langevin equations \begin{align} \dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\ \dot{y}&(t)=v_0\cos{\beta(t)},\tag{2} \end{...
Balam's user avatar
  • 11
0 votes
0 answers
84 views

What probability distribution is this?

Thank you in advance for any suggestions or feedback. I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$. I am interested in finding the Wasserstein (...
user979797987678's user avatar
1 vote
1 answer
351 views

in Euclidean space defined by multivariate normal distribution, what fraction of points falls within n-ball (centered at origin) tangent to point p?

In a Euclidean space defined by the multivariate normal distribution, what fraction of all points falls within or are tangent to (as opposed to falling outside of) the n-sphere whose center is at the ...
aputnamist2's user avatar
0 votes
0 answers
113 views

How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?

I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another. Could you please ...
Penelope Benenati's user avatar
0 votes
1 answer
207 views

Distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Cauchy

Suppose that $X$ and $Y$ are Cauchy-distributed with $\gamma=1$, i.e., with PDF $\frac 1 \pi \frac 1 {1+x^2}$. I tried to find the distribution of $R = \sqrt{X^2+Y^2}$. The PDF of $R$ should be given ...
FusRoDah's user avatar
  • 3,680
2 votes
1 answer
420 views

Is total variation distance of normalized sum of random variables to Gaussian monotonic decreasing?

Let $X_1, X_2, \ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$ and let $S_n = (X_1 + \cdots + X_n)/\sqrt{n}$ to be their normalized sum. Define $D_n$ ...
Sandeep Silwal's user avatar
1 vote
1 answer
266 views

Uniqueness of deconvolution after convolution?

I have the following question and I'd greatly appreciate any help! Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$ ...
Chang Kevin's user avatar
3 votes
1 answer
303 views

Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)

If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
Vincent Granville's user avatar
2 votes
0 answers
162 views

Random sets of points and hyperplanes in high dimensions

We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \in\mathbb{R}^d$ selected uniformly at random from the unit origin-centered ball $\mathcal{B}^{d}$. Consider the random ...
Penelope Benenati's user avatar
3 votes
0 answers
92 views

Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix

In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
Tom's user avatar
  • 279
1 vote
1 answer
243 views

Upper bound for $P(X \geq x)$, where $X \sim \operatorname{Pois}(\lambda)$

I posted the following question in a comment on CDF of a log-concave discrete random variable. Since it is not related to my main question, I thought of reposting it as separate post. Question: Let $X ...
SL_MathGuy's user avatar
2 votes
1 answer
327 views

Covariance/Correlation matrix of $n$ random variables with uniform marginal distributions

Let $X_1, \cdots, X_n \sim \mathrm{Unif}[0,1]$ be $n$ random variables, each with marginal distribution being a standard uniform distribution. I want to characterize the set of covariance matrices (or ...
zxzx179's user avatar
  • 205
0 votes
1 answer
223 views

CDF of a log-concave discrete random variable

In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave. My questions: What can we say about this in the discrete setting?. For ex: Is the ...
SL_MathGuy's user avatar
1 vote
1 answer
92 views

Geometric sampling problem in the Euclidean space in high dimensions

Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$. Question: What computationally efficient strategy can we use to sample a point $\...
Penelope Benenati's user avatar
22 votes
7 answers
5k views

What makes Gaussian distributions special?

I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions. ...
5 votes
1 answer
261 views

Random domino tilings: Is this distribution well-defined, and how can it be sampled from?

I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means. My first instinct was to do ...
RavenclawPrefect's user avatar
2 votes
0 answers
81 views

Concentration inequalities for sets

Assume that we have a random set $B$ which is constructed by selecting elements from $U = \{ X_1, \dots, X_n \}$ where $X_i$ are independent samples from Gaussians with means $\mu_i$ and variances $\...
bolzano's user avatar
  • 143
0 votes
1 answer
116 views

Existence of sequence of distributions

This question concerns distributions $\mu$ over the naturals $\mathbb{N}=\{1,2,\ldots\}$. For $q\ge1$, let us define the $q$th moment of entropy: $$ H_q(\mu)=\sum_{i=1}^\infty \mu(i)|\log\mu(i)|^q, $$ ...
Aryeh Kontorovich's user avatar

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