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.
1,921
questions
6
votes
3
answers
1k
views
Integral of product of gaussian CDF and PDF
Looking for an analytic solution to the integral below:
$$
\int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx
$$
where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
4
votes
1
answer
195
views
Uniform iid sequence
The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?
1
vote
1
answer
148
views
Joint pdf of N generally correlated (absolute values of) R.Vs as a joint pdf of (absolute value squared) random variables
The joint pdf of $|g_1|,|g_2|,\ldots,|g_N|$ is given by
$P_{|g_1|,|g_2|,\ldots\ldots,|g_N|}(r_1,r_2,\ldots,r_N)=$ $$\prod_{
\substack{k=1 \\ \mu_1 \triangleq 0}}^N \frac{2r_k}{\sigma^2(1-\mu_k^2)} \...
0
votes
1
answer
205
views
Orthogonal polynomials w.r.t. an arbitrary measure
Consider a random scalar variable $X$ with arbitrary measure.
I'm after a basis of polynomial functions $\{p_k\}_{k=0}^\infty$ which are orthonormal with respect to $X$ in the sense that
\begin{...
12
votes
2
answers
520
views
Generating function for counting partitions with corners
A corner of an integer partition is a location at where a box can be added to its Ferrers diagram to give a new partition.
E.g. the partition $\{1,1,1\}$ has two corners, and $\{1,2\}$ has three ...
1
vote
0
answers
75
views
Minima of a cdf of multivariate normal distribution with respect to a parameter
Let $\mathrm{X}\sim\mathcal{N}_{3}(\boldsymbol{\mu},\mathrm{\Sigma})$ where
\begin{equation}
\boldsymbol{\mu} = n[(\mu_1-\mu_2)\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)}, (\mu_1-\mu_3)\sqrt{\xi_1\xi_3/(\xi_1+\...
3
votes
0
answers
278
views
Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix
Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
3
votes
1
answer
260
views
Discrimination between set of binary distributions
Suppose we know two sets of distributions
$A=\{p_1,p_2,\cdots,p_k\}$ and $B=\{q_1,q_2,\cdots,q_k\}$.
We are given $C=\{r_1,r_2,\cdots,r_k\}$ such that $r_i=p_i$ for all $i$ or $r_i=q_i$ for all $i$.
...
1
vote
0
answers
89
views
If all moment of X are greater than all moment of Y, can we said something about their probability?
Consider a continuous probability distribution $f$ and two random variables $X, Y$ both are greater than equal to $0$ and they are not identical random variables.
Let's say one can show that $E[X]^k \...
1
vote
0
answers
51
views
What can we say about $\mathbb{E}[\mathrm{tr}A^{1/2}]$ for $A=\frac{1}{C}\sum_{i=1}^\infty c_i \alpha_i\alpha_i^\top \in\mathbb{R}^{m\times m}$?
Suppose we are given a summable sequence $(c_i)_{i\in\mathbb{N}}$ with $\sum_{i=1}^\infty c_i = C<\infty$ and independent $m$-dimensional, standard Gaussian vectors $\alpha_i\sim\mathcal{N}(0,I_m)$,...
4
votes
1
answer
251
views
Bounds on discrepancy metric of product measures
Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces
$$X_1^{q} = (\times_{i=1}^q\...
0
votes
0
answers
49
views
Inclusion-exclusion in a set of multivariables
I want to determine the risk of a multi-asset portfolio, in a way different from previous attempts. It is because current methods focus on just the correlation coefficient of two variables, while in a ...
1
vote
2
answers
208
views
Connection between invariant measure and positive recurrence for continuum state space markov chain
Let $\{ X_n(\omega,x)\}_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ...
9
votes
1
answer
451
views
What do category theorists know about "probabilistic metric spaces"?
I recently stumbled upon the notion of probabilistic metric space as a generalization of Lawvere's metric spaces, and I am very interested in understanding it deeper.
In short, instead of a distance $...
1
vote
1
answer
194
views
Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
2
votes
0
answers
119
views
Optimal Monte Carlo Trace Estimator
For a psd real symmetric $d\times d$ matrix $A$ and a function $f: \mathbb{R}^d \to \mathbb{R}$, with $f(x) := x^T A x$ we have that with $p(x) = \mathcal{N}(0_d, I_d)$ (i.e. standard multivariate ...
3
votes
1
answer
417
views
An inequality relating the Kullback-Leibler divergence of two discrete distributions with constant reference distribution
Suppose that $D_{KL}(p_1\parallel q)<1$ and $D_{KL}(p_2\parallel q)<1$. I'm trying to show that either $D_{KL}(p_1\parallel p_2)$ or $D_{KL}(p_2\parallel p_1)$ will have an upper bound close to ...
3
votes
2
answers
508
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 ...
0
votes
1
answer
269
views
When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
3
votes
1
answer
439
views
$\cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\frac{x}{27})\dots$ as $x \rightarrow \infty$
It is known that
$$
\cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O_{x \rightarrow \infty}(x^{-1})
$$
Is it true that
$$
f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\...
1
vote
1
answer
205
views
Using Hoeffding inequality for risk / loss function
I've got a question to the Hoeffding Inequality which states, that for data points $X_1, \dots, X_n \in X$, which are i.i.d. according to a probability measure $P$ on $X$, we find an upper bound for:
$...
2
votes
1
answer
357
views
Self normalized sum of products of i.i.d. random variables
Let $p\in (0,1)$ and $X_1, X_2, ...X_n \sim \text{Bern}(p)$ be $n$ i.i.d. Bernoulli random variables, where the probability that $X_i$ is $1$ equals $p$.
Fix $a,b>0$ different from $1$ that satisfy ...
1
vote
1
answer
116
views
Expectation value of random GUE matrix
Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors.
I wonder if one can compute (or at least get a non-trivial lower bound on) the expectation value
$$\...
3
votes
0
answers
82
views
Is the Beta distribution decomposable?
Consider a random variable $Y$ that follows the Beta distribution with parameters $\alpha$ and $\beta$:
$$
Y \sim {\rm Beta}(x | \alpha,\beta)
$$
Is it possible to decompose $Y$ as
$$
Y = X_1 + X_2
$$
...
1
vote
0
answers
48
views
Characterizing set of IID average of symmetric positive semidefinite matrices matrices
Let $\mathcal{S}_+^d$ denote the family of real $d \times d$ symmetric (strictly) positive definite matrices.
Define $\mathcal{P}_d$ to be those measures $\nu$ on $\mathcal{S}_+^d$ (assumed to have ...
1
vote
1
answer
80
views
Compressing covariance matrix from 3X3 to 2X2
thanks for answering my question. My question is,
Let $v_3=[a,b,c]^T$ is a probabilistic 3-D vector variable which is distributed by zero mean Gaussian of dense covariance matrix $\Sigma_{3\times3}$.
...
2
votes
0
answers
91
views
Approximating a probability density with a point set
Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
&...
1
vote
1
answer
90
views
Estimating the variance of Monte Carlo estimators for $F_Z$ and $f_Z$, $Z=X/Y$
This question has been migrated from the MSE.
Background/Motivation:
We have $Z=X/Y$ where $X$ and $Y$ are independent and $X\sim\mathcal N(\mu,\sigma^2)$. The density of $Y$ is not important here. ...
2
votes
1
answer
291
views
Law of large numbers for triangular arrays whose moments "look independent"
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\...
3
votes
0
answers
107
views
$\epsilon$-net for the set of distributions
Let $S$ be a space of all distributions over a finite set $X$. What is the size of its $\epsilon$-net w.r.t. the total variation distance defined as $d(P,Q) = \frac12 \sum_{x \in X} |\Pr[P=x] - \Pr[Q=...
0
votes
0
answers
50
views
Using random matrix theory to calculate Lyapunov spectrum; relation between cumulative distribution and the spectrum
I am reading a paper using random matrix theory to calculate Lyapunov spectrum.
What particularly confuses me is
Why is the Lyapunov spectrum simply the inverse of $\chi(x)$ (the probabilistic ...
0
votes
1
answer
340
views
Fokker-Planck: uniqueness and convergence to stationary distribution
Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
1
vote
1
answer
230
views
Bounding $2$-Wasserstein distance and the $L^1$ distance
My questions come from the paper Logarithmic Sobolev inequalities for some
nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
1
vote
2
answers
119
views
Existence of a strictly proper scoring rule on a $\sigma$-algebra that is not countably generated
Given a $\sigma$-algebra $\scr F$ on $\Omega$, say that an accuracy scoring rule for $\scr F$ is a function $s$ from the set of all (countably additive) probabilities on $\scr F$ to the $\scr F$-...
7
votes
2
answers
3k
views
Expected value of min of variables - what informations do I need?
I encountered a problem where I need to compute:
$$\mathbb{E}(U) = \mathbb{E}(\min(X_1, .. , X_6))$$
The problem is that I have little information on the $X_i$. Basically I know $\mathbb{E}(X_i)$ and $...
1
vote
1
answer
94
views
Tight upper-bound on dependent events
Given $r$ r.v.s $x_k$: $x_k=1$ with probability $p_k$ and $0$ otherwise. Let $s_i=\sum_{j=1}^r c_{ij}p_j$ where $c_{ij}\in[0,1]$, $1\le i\le n$. Let $E_i$ denote the event $\sum_{j=1}^r c_{ij}x_j>(...
1
vote
0
answers
110
views
What are the Lévy processes with specific increments?
It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$.
I wonder what are the Lévy processes with increments from ...
3
votes
1
answer
93
views
On an integral of Gaussian CDFs
Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by
$$F(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$
For every $a\in [0,1)$, ...
-1
votes
2
answers
150
views
Cumulants of a sequence of variables with zero mean and variance
Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
0
votes
1
answer
163
views
How to calculate the dual spaces of the following spaces?
Let us consider the spaces $C_\infty(E)$, $C_c(E)$, $C_b(E)$, where $E$ is locally compact, $C_\infty(E)$ is all continuous functions vanishing at the ends of $E$, $C_c(E)$ is all the continuous ...
0
votes
1
answer
97
views
Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?
Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?
1
vote
0
answers
53
views
A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
3
votes
3
answers
1k
views
How close are two Gaussian random variables?
Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?
0
votes
0
answers
34
views
How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?
I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...
0
votes
0
answers
65
views
Asymptotic distribution of a specific ML estimator
Consider a random independent sample of size $n$ from a distribution defined by the following probability density function$$f(x)=\frac{3}{4\theta}\cdot\left(1-\frac{x^2}{\theta ^2}\right)$$ when $-\...
1
vote
1
answer
211
views
The maximum trace of a covariance can be achieved by a discrete random vector?
Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =...
4
votes
1
answer
451
views
Transforming a Poisson distribution into a power law
Consider the probability mass function of the Poisson distribution given a mean $\lambda$:
\begin{equation}
\mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !}
\end{equation}
By ...
0
votes
0
answers
62
views
The moment problem for $m_n=1/n$
What is the p.d.f. for the moments $m_n=1/n$ ?
(They are obtained from $\int_0^1 x^n/x\ dx $, but clearly $1/x$ is not a p.d.f. on $[0,1]$)
5
votes
2
answers
138
views
Number of resampling until obtaining a uniform list
Let $A_0$ be a list of $ n$ distinct elements. By sampling with replacement the elements of $A_0$, we obtain a new list $A_1$ of $n$ elements that are not necessarily distinct. Repeat the same process ...
8
votes
0
answers
380
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...