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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Why is it impossible to create a numerically balanced die with more than 120 sides?
I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
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1
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Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
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1
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197
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Uniqueness of the variance
The variance assigns a number to each of certain probability distributions on Borel subsets of $\mathbb R$. It has the properties of
(1) shift-invariance, i.e. if $X$ is a random variable with a ...
2
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53
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Dirichlet series solution to Poisson Point Process question (repost from math.SE)
Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received.
For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
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Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
3
votes
1
answer
354
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Relative entropy equality for a sequence of Bernoulli random variables
We are given two joint probability distributions, $p$ and $q$, of $n$ Bernoulli random variables $X_1, X_2, \ldots, X_n$.
We denote by $p(x_k\mid x^{k-1})$ the probability $\mathbb{P}_p(X_k=x_k\mid ...
3
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2
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857
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Why MLEs are asymptotically efficient whereas method of moment estimators are not?
Under appropriate regularity conditions it is well-known that Maximum Likelihood Estimation (MLE) produces asymptotically efficient estimators in the sense that their asymptotic covariance is given by ...
2
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95
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Equivalence of score function expressions in SDE-based generative modeling
I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...
0
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1
answer
127
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A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$
Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, ...
5
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2
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880
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Subgaussian norm of a symmetric $\{-1,0,1\}$ random variable
Let $p\in [0,1/2]$, and define $\xi$ as the symmetric random variable such that
$$
\xi = \begin{cases}
1 & \text{ w.p. } p\\
0 & \text{ w.p. } 1-2p\\
-1 & \text{ w.p. } p
\end{cases}
$$
so ...
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1
answer
41
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Expected value of MGIG distribution
I'm currently dealing with a Gibbs sampler of the multivariate generalized inverse Gaussian distribution (MGIG). In order to check the correctness of the sampler, I'd like to know the expected value ...
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27
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Distribution of covariance matrix of vectors of lognormal rvs
I know that the Wishart distribution is the distribution of the sample covariance matrix of a multivariate normal distribution. I am wondering if the analog distribution for a sample from a ...
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36
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Conditions for an ODE with convolution term to have a probability distribution solution
Suppose we have a simple ODE like:
$$
y''(x) + 2ay'(x) + by(x) = 0
$$
with the condition $y(0)=0$ for $x\leq 0$. Then the solution on $(0,\infty)$ will be of the form $Axe^{-ax}$ when $a^2=b$, $Ae^{-...
2
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2
answers
231
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Convergence of the row sums in a triangular null array with zero mean
Let $(X_{jn})_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the ...
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132
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An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
2
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2
answers
152
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How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
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63
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Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant
Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse.
...
4
votes
1
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185
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On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
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2
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Sum and alternating sum of a series of Bernoullian variates
Consider the random variables
$a_i,i=0,1,\ldots,n$ be random variables which take values from $\{-1,1\}$ independently and randomly with equal probability. Let
\begin{align}
S &= a_1+\cdots+a_n , ...
1
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1
answer
101
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What is this distributional closeness?
Let $P$ and $Q$ be two distributions over a sample space $\Omega$ which I would like to show are close under some choice of distance function. So far I have managed to show that there exists a subset $...
4
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1
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246
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Does a subset with small cardinality represent the whole set?
Assume that we have heavy-tailed distribution $F(x)$ such that
\begin{align}
F(x)=\mathbb{P}[X\geq x]=x^{-0.5}.
\end{align}
Then, we produce $N$ independent samples $X_1,X_2,\ldots,X_N$ from this ...
0
votes
2
answers
263
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Simplification of hypergeometric Function
First of all I am not at all a math expert, but I have some working knowledge.
That said, please excuse "dumb" questions.
I am looking at the following process: Assume you are on the 2-...
3
votes
2
answers
207
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Recovering measure support from the sequence of I.I.D random variables
Assume we have a Borel probability measure $\mu$ in $\mathbb{R}^n$ and a sequence of $\mu$ distributed I.I.D. random variables $x_n$. Is there a limit formula for $supp(\mu)$, something like closure ...
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0
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29
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Can a "discrete" argument involving a gamma mixture of Poisson distributions be adapted to work for non-integers?
PREFACE
Here I will begin with this preface that is nearly the same as what I wrote in an earlier question.
Suppose
$$
\Pr(\Lambda\in d\ell) = \frac 1{\Gamma(\alpha)}
\left( \frac\ell m\right)^{\alpha-...
4
votes
0
answers
219
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Weight transfer proof of Turán’s theorem
Turán’s theorem, which states that a $K_{p+1}$-free graph contains at most $(1-1/p)\frac{N^2}{2}$ edges, can be proven in many different ways, as pointed out, for example in M. Aigner, G. M. Ziegler, ...
5
votes
1
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384
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Bounding the variance of a truncated Gaussian random variable
Suppose $X_1, X_2, X_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that:
$$\text{Var}[X_2\mid X_2 \geq X_1 - a, X_1 \leq X_3 + b] < 1,$$
where ...
-1
votes
1
answer
131
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joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
0
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0
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87
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The best probability distribution for the game of Number Master
In the game of Number Master, the player controls a number starting with $1$ and hits the other numbers one by one on the road.
If the player hits a number smaller or equal to the current controlling ...
3
votes
1
answer
135
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Quantitative results (with formal proof) on the median approximation of Chi-squared distribution
It is well-known that Chi-squared distribution $X_n$ with degree-$n$ freedom has an approximate formula for its median as $n\left(1-\frac{2}{9n}\right)^3$. Or $(X_n/n)^{\frac{1}{3}}$ is approximately ...
2
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1
answer
124
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von Mises Distribution property
The von Mises distribution has the highest entropy for a given first circular moment. It seems that the converse is true: for a fixed entropy, the von Mises distribution has the highest first ...
4
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3
answers
376
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Gaussian approximation of the characteristic function of Rademacher sum
I am searching an uniform bound for the characteristic function of some Rademacher sum.
Specificaly I want to estimate how much the characteristic function is close to a Gaussian.
We have in general ...
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1
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156
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Under what conditions is a multivariate normal distribution exchangeable?
Consider a multivariate normal distribution. What are necessary and sufficient conditions (esp. on the covariance matrix) under which this distribution is exchangeable?
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33
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How to find lower bounds of a modified mixing time (defined below) with respect to spectral of a finite Markov chain?
I am focused on a time-homogeneous continuous-time Markov chain with a finite state space $\mathcal{X}$, whose Markov kernel is $K$ and the corresponding semigroup is $H_t=e^{-t(I-K)}$. The invariant ...
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0
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52
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A one-sided/monotone version of min/max-stable distributions -- does this have a name?
In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
2
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1
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107
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prove with a probability of at least $1/e$: $\left\|\sum_{i=1}^k a_{i} P_{i}\right\|_2 \geq\left\|P_{1}\right\|_2$ holds
Let $a_i (i \in\{1...k\})$ be $k$ IID standard Gaussian random variables, $P_i$ are $d$-dimensional constant vectors. How to prove with a probability of at least $1/e$,
$$
\left\|\sum_{i=1}^k a_{i} P_{...
3
votes
1
answer
165
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Variance lower bound for natural exponential family
Let $Q$ be a probability measure on $\mathbb{R}$. Let $$Q_h(dy) = e^{y \cdot h} Q(dy) / M(h) \quad \text{where} \quad M(h) = \int e^{y \cdot h} Q(dy)$$ defined for $h \in (-c,\infty)$ with some $c &...
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votes
1
answer
130
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Branching process with varying offspring distribution at each step
Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.
...
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Do finite exchangeable random sequences behave asymptotically independently?
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ and $(Y_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be triangular arrays of row-wise exchangeable random variables, that is for any $n$ and permutation $\...
4
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Projection of log-concave distribution on unit sphere surface
Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution.
Is there any known upper bound for the probability density function of $...
-1
votes
1
answer
68
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Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
2
votes
1
answer
112
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Sample integral points in m-Ball
The problem I have is pretty simple, however I cannot find an answer.
I need an efficient algorithm to sample integral points within an m-dimensional ball of radius r around the origin (Euclidean norm)...
0
votes
1
answer
85
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Is it reasonable to consider the subgaussian property of the logarithm of the Gaussian pdf?
Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y_1, Y_2, \...
2
votes
1
answer
159
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Hoeffding's Lemma for bounded complex random variables?
If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality:
\begin{align}
\mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\...
5
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1
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886
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How can I sample uniformly from a citrus surface?
I want to sample from a Lemon surface uniformly. The equation of this surface is
$$16(x^2+z^2)+(y-2)^3 y^3=0.$$
I have read the paper
Stratified Sampling of 2-Manifolds
. The method described in this ...
2
votes
1
answer
248
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Simple anticoncentration bound for binomially distributed variable
The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...
0
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0
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70
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Calculation of the difference of two Brownian bridges
I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e.,
$$B_{1t} - B_{2t} = \sqrt{2}B_t$$
where $B_{1t}$ and $B_{2t}$ are ...
14
votes
1
answer
385
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Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
0
votes
0
answers
66
views
Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$X_t=\begin{cases}
\frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\
\frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
1
vote
1
answer
181
views
$L_1$ norm concentration of an empirical distribution
Suppose we have one random variable $X$, whose sample space is $\mathbb{X}=\{x_1,x_2,\dots,x_m\}$, and the size of the sample space is $m$. We have $N$ i.i.d. samples from this distribution, and $x_i$ ...
4
votes
1
answer
417
views
An exercise on log-concave random variable on the real line
Let $X$ be a real random variable with log-concave density $f$. Assume that $E(X) =0$ and $E(X^2)=1$.
Show that there is a universal (independent of $X$) constant $c>0$ such that:
$$P(X\in[-1/2;0])\...