All Questions
Tagged with pr.probability probability-distributions
1,384 questions
2
votes
2
answers
2k
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Multivariate power law distributions?
Is there a text books or publications that describes multivariate power law/pareto distributions?
- 63.9k
1
vote
1
answer
86
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Is the main part of certain exponential family sub-Gaussian?
$X$ is in the form of exponential family i.e.
$$\mathbb{P_\theta}x = h(x)e^{\langle \theta,T(x)\rangle-\phi(\theta)}$$
where $\theta\in \mathbb{R}^d$. If $\nabla\phi(\theta)$ is L-Lipschitz i.e.
$$\...
- 128k
2
votes
1
answer
324
views
On the mean value taken by Bernoulli random variables with joint distribution constraints
We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
- 1,677
2
votes
1
answer
415
views
Bounding Kullback-Leibler
Suppose we have a probability distribution $P$ on a finite set $S$. We draw $N$ i.i.d. samples according to $P$ and use these samples to define an empirical distribution $R$. We measure the Kullback-...
- 2,183
3
votes
0
answers
141
views
Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 620
0
votes
1
answer
161
views
Analogues of Kac-Bernstein characterisation theorem for non-normal distributions
Let $X,Y$ be two independent random variables.
The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal.
Are there analogues of this theorem for non-normal, ...
- 128k
1
vote
1
answer
81
views
Let $\alpha\in(0,1),d\in\mathbb N^+$ and $X,Y\in\mathbb S^d$ be uniform, what is $\Pr[\lVert X-Y\cdot\sqrt{1-\alpha} \rVert^2\le \alpha]$?
Suppose that $X,Y$ are independent random $d$-dimensional vectors each uniformly distributed on the unit sphere, and let $Z=Y\cdot\sqrt{1-\alpha}$ be a uniformly selected vector on a slightly smaller ...
- 128k
31
votes
5
answers
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On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1$, if $u_1$ is in $(0,1)$ and $u_k$ is in $(0,eu_{k-1})$?
A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$.
Let's tweak this question by making each random ...
- 61.9k
3
votes
1
answer
159
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Are there any known results on the probability distributions of perpetuities with power law discount rates?
Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$
where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
- 5,474
3
votes
2
answers
206
views
Getting Wasserstein closeness from a derivative estimate
In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate:
$$
|\mathbb{E}_{\mu}(f)-\...
- 2,772
0
votes
1
answer
124
views
What is the probability space corresponding to the probability measure $\mathbb{P}_{p}$ in the context of this paper?
Here is the definition of the frog model we are interested in:
"... consider the homogeneous tree $\mathbb{T}_{d}$, that is, the rooted tree in which each
vertex has (is connected by edges to) $d ...
- 3,937
0
votes
1
answer
102
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How does this Bayesian updating work $z_i=f+a_i+\epsilon_i$
$z_i=f+a_i+\epsilon_i$ ，where $f\sim N(\bar{f},\sigma_{f}^2)$ ; $a_i\sim N(\bar{a_{i}},\sigma_{a}^2)$; $\epsilon_i\sim N(0,\sigma_{\epsilon}^2)$. We can see the signals $\{z_i\}$ where $i\subseteq {1,...
- 211
4
votes
1
answer
2k
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Examples of convergence in distribution not implying convergence in moments
It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions.
Let $\{X_n\}$ be a ...
- 869
5
votes
0
answers
96
views
Is there a name for the set of distributions whose probability generating functions are Mobius transformations?
Consider a discrete random variable $N\in\mathbb N$ with
$\mathbb P(N=0) = p$,
$\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$.
Then the probability generating function of $N$
$$\mathbb E(z^N) = \...
- 2,821
1
vote
1
answer
195
views
CDF of sum of independent cosines?
Consider the random variable
$$X=\frac{1}{d}\sum_{k=1}^d\cos X_k$$
where $X_k$ are each drawn uniformly i.i.d. from $[0,2\pi]$. What is the CDF of X?
It seems that a relatively direct way could be to ...
- 188k
4
votes
1
answer
236
views
Expected p-norm of binary vector
Let $\sigma=(\sigma_1,...,\sigma_m)$ be i.i.d. uniform binary 0-1 valued variables.
I'm trying to figure out what is the order of $E[||\sigma||_p]$ with respect to $m$.
Jensen's inequality gives an ...
- 128k
5
votes
1
answer
613
views
Radon-Nikodym derivative and conditional probability
In this paper by Diaconis and Zabell from 1982, Theorem 2.1 and the remark after essentially stated that
Given two probability measures $P$ and $Q$ on the same probability space $\Omega$. If $Q\ll P$ ...
- 17k
0
votes
2
answers
101
views
Minimal set of functions to characterize a distribution
In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
- 18.7k
6
votes
1
answer
894
views
Expected value of orthogonal projection $X^{+}X$
Let $X\in\mathbb{R}^{m\times n}$, where $m<n$, be a random matrix where the rows $x_i$ ($i=1,...,m$) are sampled i.i.d. from Gaussian distribution with mean $0$ and covariance $\Sigma$, i.e. $x_i\...
- 1
3
votes
2
answers
505
views
Precise asymptotics for moments of order statistics of normal distribution
Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
- 128k
2
votes
0
answers
201
views
Continuity of density of SDE
Consider a stochastic differential equation in $\mathbb R^m$ with a parameter $\theta\in\mathbb R$:
\begin{equation}
dX_t^{\theta,x} = v(\theta,X_t^{\theta,x})dt+\sigma(X_t^{\theta,x})\circ dW_t,~...
- 6,215
0
votes
1
answer
188
views
Equality cases in a certain case of Jensen's inequality
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when
$$...
- 128k
1
vote
1
answer
97
views
A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable
Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
To get the non-strict version of ...
- 36
1
vote
2
answers
570
views
An inequality on Difference of Entropies
Hi,
I have the following problem that came up. It is not a homework problem or something similar. I did my simulations and it seems to hold but i was unable to prove it.## Heading ##
Let $P$ and $Q$ ...
- 2,821
1
vote
1
answer
351
views
Is the Wasserstein distance to the empirical measure minimized by the underlying distribution?
Let $S$ be a metric space and denote the set of probability measures on $S$ by $\mathcal{P}(S)$. Fix $\mu\in \mathcal{P}(S)$ and denote the law of $N\geq 1$ i.i.d samples $X=(X_1,\ldots,X_N)$ from $\...
- 128k
0
votes
1
answer
272
views
Maximal mutual information between a continuous and a discrete random variables
Let $X\sim \mathcal{N}(\mu,\sigma^2)$ be a Gaussian random variable with random mean $\mu\sim {\sf Bernoulli}(p)$, i.e., $\mu=1$ with probability $p$ and $\mu=0$ with probability $1-p$. In other words,...
- 128k
7
votes
2
answers
392
views
On a von Bahr–Esseen-type inequality for pairwise independent zero-mean random variables
For $p\in(1,2)$, let $C_p$ be the smallest constant factor $C$ in the von Bahr–Esseen-type inequality
\begin{equation}\label{eq:pair}\tag{1}
E\Bigl\lvert\sum_{j=1}^n X_j\Bigr\rvert^p\le C\sum_{j=1}...
- 128k
3
votes
0
answers
179
views
Probability terminology
This is strictly a low-level terminology question. If I have a probability space $\Omega$ and a measurable space $S$, then a random variable $X:\Omega\rightarrow S$ gives rise via pushforward to a ...
- 63.9k
0
votes
0
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128
views
When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?
In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...
- 420
1
vote
1
answer
82
views
Does the following expectation-based inequality hold?
Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...
- 128k
0
votes
1
answer
115
views
Order of orthant probabilities in a prolate multinormal distribution
This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...
- 4,219
6
votes
1
answer
264
views
Which orthant probabilities are the largest? (For a multivariate normal distribution)
I have a $k$-dimensional multivariate normal distribution $X∼N(0,\Sigma)$ with covariance matrix $\Sigma$. $\Sigma$ has two distinct eigenvalues, say $\lambda_1 > \lambda_2$, with orthogonal ...
- 4,219
1
vote
0
answers
276
views
Lipschitz function of a sub Gaussian vector
I have been struggling with the following question.
Let $X \in \mathbb{R}^n$ be a $K$ sub Gaussian random vector (i.e. $\|\langle u, X \rangle\|_{\psi_2} \leq K$ for all $\|u\|=1$) and let $f : \...
- 11
0
votes
1
answer
218
views
Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV?
Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?
- 284
6
votes
3
answers
2k
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Estimating the variance of a discrete normal distribution
Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
- 63.9k
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 61.9k
5
votes
3
answers
512
views
Optimisation under constraint of Wasserstein distance
Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or ...
1
vote
1
answer
75
views
Does bounding mutual information restrict the defined meter?
Suppose $I(X;Y)$ denotes mutual information and on the other hand there is a relationship as follows.
\begin{align}
|p(y)-p(y|x)|<\delta p(y),\qquad\forall x,y.
\end{align}
Then we can say about ...
- 128k
3
votes
1
answer
171
views
Exponential of supremum of Brownian bridge on short time frame
For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$.
Question: Is it true that $\mathbb E[|\text{exp}\, (\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $...
- 128k
6
votes
1
answer
248
views
Violating an order statistic inequality?
[Edit: for posterity, I'm adding two small comments to the code explaining how to fix it, in light of Iosef Pinelis' answer below. Look for "Should be:" to find the corrections.]
Suppose we ...
- 3,979
2
votes
1
answer
188
views
Probability distribution of vectors obtained from Gram-Schmidt process on i.i.d. Gaussian vectors
Given $N$ vectors in $K$ dimensions that are independently and identically distributed according to a Gaussian distribution with mean $0$ and standard deviation equal to an identity matrix, what is ...
- 188k
11
votes
2
answers
1k
views
Distribution of infinity-norm over the unit sphere
I need to compute probabilities of the form
$P( \Vert X \Vert_\infty < r ),$
where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere $\mathcal{S}_{n-1}$...
- 4,713
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 321
0
votes
1
answer
169
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Understanding the approximation of a random sum of random processes
I want to understand an approximation of a compound Poisson distribution in this paper.
First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly ...
- 128k
0
votes
1
answer
159
views
Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array
We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $(X_{jn})_{1\leq j \leq n}$ be a i.i.d. triangular array such that
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...
- 128k
2
votes
1
answer
139
views
Stochastic inverse
Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying
$$
\int_0^Tf(H_t,...
- 5,474
2
votes
1
answer
1k
views
Bootstrapping and the central limit theorem
I have been looking into bootstrapping lately and although I believe to have understood the basic process somewhat, I am fuzzy on the mathematical details. I will begin with my understanding of what ...
- 128k
5
votes
3
answers
665
views
The relative error of approximating a binomial
Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...
- 37.7k
1
vote
0
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43
views
Does the constrained Wasserstein barycenter admit a blue noise property?
Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
- 167
3
votes
2
answers
667
views
Is every discrete compound Poisson distribution a mixed Poisson distribution?
I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer.
The mixed Poisson distribution and compound Poisson ...
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