All Questions
Tagged with pr.probability probability-distributions
1,386 questions
2
votes
1
answer
210
views
Controlling Mean Difference Between Product and Joint Distributions Using Optimal Transportation
Suppose we have nonindependent random variables $X \sim P$ and $Y \sim Q$, where $P$ and $Q$ denote their marginal distributions. We are interested in upper bounding
$$
|\mathbf{E}_{X, Y\sim P \...
- 128k
2
votes
2
answers
1k
views
Uniform correlation matrix sampling and not so uniform laws
Hi everyone,
I am looking for a way of simulating correlation matrices of fixed dimension in (at least) two ways.
First, I would like to determine the "uniform" distribution over the "correlation ...
- 1,334
3
votes
1
answer
209
views
Log concavity of the maximum of dependent Gaussians
Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
- 128k
0
votes
1
answer
270
views
Bivariate Poisson-Binomial distribution
Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...
- 173
0
votes
1
answer
731
views
Is the normal product distribution sub-gaussian?
Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
- 29.8k
3
votes
1
answer
178
views
Tail probability of random projection
Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
- 128k
2
votes
0
answers
302
views
Schilder's theorem for brownian bridges
I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.
A bit of context: usually, Schilder's theorem tells us that the ...
- 5,380
2
votes
1
answer
258
views
A modest generalization of the law of large numbers
Suppose I collect $2n$ independent samples of a probability density function $f$, which are separated into pairs $\{X_i^1, X_i^2\}$ for $1\leq i\leq n$. Suppose I now consider the set of all $2^n$ ...
- 23.2k
4
votes
1
answer
649
views
What is the probability for a Binomial to be greater than other?
Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$.
What is the probability for $X$ ...
CommunityBot
- 1
2
votes
1
answer
171
views
Stochastic domination of Gaussian random vectors
Let $S$ be the class of all $2$ by $2$ matrices of the form
$$\begin{bmatrix}
1 & a \\
a & 1
\end{bmatrix},\, |a|\leq 1.$$
Is there a single matrix $M\in S$ such that for ...
- 5,721
4
votes
1
answer
580
views
Tracking down locality assumption in CHSH inequality
CHSH inequality requires both locality and realism. I will equate here realism with counterfactual definiteness.
Now counterfactual definiteness tells us that given two different measurements on the ...
- 103
3
votes
1
answer
167
views
Lower bound on the sum of pmf squared of a hypergeometric distribution
I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
- 33
4
votes
1
answer
2k
views
wasserstein distance between distributions with bounded ratio
Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying
$$
\alpha d p \le dq \...
- 6,853
9
votes
0
answers
389
views
How do I draw samples from this distribution?
Let S be the the standard K-1 simplex. Consider the following probability distribution:
$$\begin{align}
f(p,\alpha,\beta) &= \prod_{k=1}^K p_k^{\alpha_k-1}(1-p_k)^{\beta_k-1}\\
Z(\alpha,\beta) &...
CommunityBot
- 1
13
votes
1
answer
10k
views
KL divergence and mixture of Gaussians
Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good ...
- 188k
2
votes
2
answers
104
views
Draw samples from distribitions in the neighborhood of a fixed distribution
Disclaimer
Sorry in advance for vagueness. I'm still trying to get my ideas right on this one.
Setup
So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...
- 1,095
2
votes
1
answer
2k
views
How to estimate a total variation distance?
Let $X_1, \ldots, X_n$ be independent Bernoulli random variables. Then $Pr[X_i=1]=Pr[X_i=0]=1/2$. Let $X = (X_1, \ldots, X_n)$ and $v \in \{0,1\}^n$, $Y=v \cdot X$, $Z=Y-1$. Let
\begin{align}
\mu_1(x)...
- 128k
14
votes
1
answer
4k
views
Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension
By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
- 128k
7
votes
2
answers
460
views
Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws
It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...
CommunityBot
- 1
3
votes
1
answer
129
views
Reference Request: Simple Random Walk on $\mathbb Z$ is Unimodal
I am looking for a reference to the following claim. Let $X = (X_t)_{t\ge0}$ be a continuous time simple random walk. Then
$$
m \mapsto P(|X_t| = m) : \mathbb N \to [0,1]
$$
is (weakly) decreasing (or ...
- 560
1
vote
1
answer
215
views
are there measure preserving mapping in this case?
Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], ...
- 41.1k
1
vote
0
answers
237
views
CLT for random sums: Anscombe's Theorem vs. "classical" version
Given a compound Poisson distribution
$$S(t):=\sum_{k=1}^{N(t)} X_{k}$$ with
$N(t)\in\mathbb{N},\,t\geq0$ a Poisson process with rate $\lambda.$
$X_{k}\in L^{2}$ are iid random variables, i.e. $\...
- 203
4
votes
1
answer
125
views
How to find the optimal convergence rate?
I have already asked that Question on Cross Validated:
Link
Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do ...
- 128k
3
votes
2
answers
278
views
The disintegration of the convolution of two probability measures
Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
CommunityBot
- 1
2
votes
2
answers
144
views
Spectrum of finite-band random matrices?
Let
$X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :
$$ \begin{cases}
&X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\
& X_{ij} \sim P_X \quad \text{otherwise}
\end{cases}$$
And ...
- 245
3
votes
0
answers
243
views
Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...
- 6,853
1
vote
0
answers
87
views
Conditonal convergence implies convergence?
Note : All measures below are probability measures.
Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$.
Actually,...
- 245
3
votes
1
answer
1k
views
Minimizing the expectation of a functional of probability distribution subject to an entropy constraint
Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the ...
- 61.9k
3
votes
1
answer
228
views
Density of a somewhat random set
The density of a set
$X\subseteq\omega$ refers to:
$\limsup\limits_{n\rightarrow\infty}\dfrac{C\cap n}{n}$.
Given a set of positive integers
$F= \{m_0<\cdots<m_{k-1}\}$,
let $C\subseteq \omega$...
- 30.1k
0
votes
1
answer
213
views
Show that interval of maximum probability grows no faster than $\sqrt{n}$ for binomial distribution
Let $X \sim \text{Binom}(n, p)$ a binomial random variable. I want to show that : $$\forall 0 < t < 0.9, \quad \exists C, \quad \forall n >1, \quad \mathbb P\bigg(|X-np| \leq C\sqrt{n}\bigg) \...
- 128k
2
votes
1
answer
628
views
Fastest convergence of sum of uniform independent distributions to a Gaussian
The sum of uniform i.i.d. random variables follows the Irwin-Hall distribution. Through observation it seems that the convergence is faster in comparison to the sum of uniform independent but not ...
- 128k
4
votes
1
answer
1k
views
Bound for a conditional expectation
Let $a_i, i=1, \ldots, n$ be real numbers. Let $\epsilon_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be random variables ...
- 34.7k
2
votes
1
answer
64
views
Maximum Number of modes of $V=U+Z$ where $Z$ standard normal and $|U|\le a$
Let $f_V$ be a pdf of random variable $V$ where
\begin{align}
V=U+Z
\end{align}
and where $U$ and $Z$ are independent and $Z$ is Gaussian. Moreover, suppose that $|U| \le A$.
Can we find the upper ...
- 128k
1
vote
1
answer
499
views
property of iid random variable
Let $ (\xi_i)_{i \ge 1} $ be independent identically distributed random variables, taking values in $ (1,3]$.
Can we show:
$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{...
- 128k
4
votes
1
answer
2k
views
Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?
I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...
- 141
1
vote
1
answer
372
views
Calculate Average and Correlation of WSS Random Processes
Given two stochastic processes, $X[n]$ and $Y[n]$, both being WSS (wide state stationary) and independents. What would be the Average and Autocorrelation function of $Z[n] = Y[n] X[n]$?
Is the ...
CommunityBot
- 1
9
votes
1
answer
180
views
Variant of mutual information
Given a discrete random variable $(X,Y)$, one can consider the smallest entropy of a random variable $Z$ such that $X$ and $Y$ are independent conditioned to $Z$.
This quantity is akin to the mutual ...
- 76
0
votes
1
answer
189
views
Why Expected squared length of a projected vector on reduced dimensionality coordinates is k/d?
For the proof of Johnson-Lindenstrauss algorithm by Dasgupta and Gupta, they comment in their paper that for a vector $Z \in R^k $, the expected squared length, $E[L]$ (where $L = \|Z\|^2$) of the ...
- 3,574
33
votes
1
answer
2k
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 ...
- 128k
2
votes
1
answer
2k
views
Explicitly representing a random variable in terms of indicator functions
Motivation:
I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula.
I want to prove the change of variable formula (you ...
- 247
10
votes
0
answers
742
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 "...
- 13.6k
3
votes
0
answers
116
views
Trace of Symmetric matrices in fixed rank
I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem:
For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...
- 179
5
votes
1
answer
942
views
Moments of maximum of independent Gaussian random variables
Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
- 128k
1
vote
0
answers
109
views
Bounding quantiles of the noncentral chi distribution
I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
- 162
2
votes
1
answer
689
views
One question about compensated Poisson process
Let $N$ be a Poisson process with parameter $\lambda$, that is, for $a>b\geq0$, there is $$P[N(a,b)=k]=\frac{((a-b)\lambda)^k}{k!}e^{-(a-b)\lambda}.$$ Now denote $N_t=N[0,t)$ and define
$$
M_t=N_t-...
- 128k
1
vote
0
answers
67
views
Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
- 19
1
vote
1
answer
480
views
Ratio of perfectly correlated gaussian distributions
Let $M$ be a positive definite matrix and let $w \in S^{d-1}$ be a unit vector uniformly distributed over the sphere. I want to understand the distribution of the quadratic form $\frac{w^T M^3 w}{w^T ...
- 128k
4
votes
0
answers
100
views
Asymptotics of the joint pdf of two sums of powers of independent $\mathcal U(0,1)$ random variables
As a warm-up in words: The sum of twelve uniform random variables is a classic approximation to a normal distribution. What is the joint pdf for the sum of their cubes and the sum of their fourth ...
CommunityBot
- 1
2
votes
1
answer
1k
views
$p$-th moment of complex Gaussian random variable
Let $1<p<2.$ Let $G$ be a complex Gaussian random variable. then what is the value of $\mathbb{E}[|G|^p]$ ? The symbol $\mathbb{E}$ denotes the expectation of a random variable.
- 128k
0
votes
1
answer
308
views
Berry-Esseen type theorem for Monotonic independence
The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
- 4,034