All Questions
Tagged with pr.probability probability-distributions
1,384 questions
6
votes
1
answer
481
views
Probabilistic Proofs of Key Number-Theoretic Results
Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$.
Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
- 329
1
vote
1
answer
517
views
log-like distance between probability distributions
Given two probability density functions (PDF) $f$ and $g$, both defined over the same set $X$, there are many ways to describe/measure the distance between them, e.g., KL divergence and Hellinger ...
- 3,574
4
votes
2
answers
108
views
A density function that matches the $k$ smallest elements of $n$ uniform samples
Let me apologize in advance as this feels like a homework question, though I've tried without success to work through the relevant integrals. My question is: given positive integers $k$ and $n$ with ...
- 4,049
7
votes
3
answers
278
views
Solve $\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] $
I want to solve the following optimization problem
\begin{align}
\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right]
\end{align}
where $X^\prime$ is an independent copy ...
- 671
1
vote
1
answer
231
views
Are interarrival times of doubly-stochastic Poisson I.I.D.?
I am working on a Markov-modulated Poisson process $\{N_{t}, t \geq 0\}$, which is itself a Poisson but the rates of which are governed by a CTMC. In my case, the CTMC is a one-class, aperiodic and ...
- 48
2
votes
0
answers
247
views
Moments of a Normal-Wishart distribution
Do known expressions exist for the moments of a gaussian-wishart (aka normal wishart) distribution?
$$NW(\mu,K\mid\mu_0,\lambda_0, v, W) =
\frac{|\lambda_0K|^{1/2}}{(2\pi)^{d/2}}e^{-0.5([\mu - \mu_0]...
- 121
8
votes
3
answers
8k
views
Upper bound total variation by Wasserstein distance for continuous distance
I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper).
The general results show that for general distributions, we ...
- 163
3
votes
2
answers
430
views
Multivariate normal concentration
If $X\sim N(0,\Sigma)$ for some $d$-dimensional normal distribution, then $X = \Sigma^{1/2} Z$ where $Z\sim (0,I)$. How to compute the following quantity?
$$
\operatorname{var} (X^T X)
=
\...
- 719
5
votes
0
answers
523
views
How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently
I hope you are well. Here is my problem.
Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...
- 173
3
votes
1
answer
281
views
Stein's Equation for Gaussian Mixtures
In the paper "Spin glasses and Stein's method" (https://arxiv.org/pdf/0706.3500.pdf), Sourav Chatterjee established Stein's equation for mixtures of two Gaussian densities in $\mathbb{R}$, which takes ...
- 1,127
5
votes
3
answers
117
views
Looking for a certain kind of a distribution
Is there any probability distribution supported on a compact or a half-open interval (of $\mathbb{R}$) such that if a vector $\vec{x} \in \mathbb{R}^n$ is sampled by sampling its coordinates like that ...
- 2,246
1
vote
0
answers
386
views
Bounds on the distance between probability distributions in terms characteristic functions
I am looking for the bounds on the distance between probability distributions in terms characteristic functions.
For example, I am aware of the following bound
\begin{align}
d(P,Q) \le \frac{1}{T} \...
- 671
4
votes
0
answers
91
views
What is the entropy of binomial decay?
Let's play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this ...
- 141
2
votes
0
answers
191
views
Splitting of a random variable
Can we characterize the laws of n-uple real random variables $(X_1...X_n)$ so that for any random variable Y which has the same law as the sum of the $X_i$ there exists a n tuple of functions $(f_i)_i$...
- 41
6
votes
2
answers
2k
views
Is there a universal bound for this ratio of expectations?
Let $X$ and $Y$ be two zero-mean independent and identically distributed random variables. Is there a bound for the following ratio,
$$\frac{\mathbb{E}[|X+Y|]}{\mathbb{E}[|X|+|Y|]}=\frac{\mathbb{E}[|...
- 287
1
vote
0
answers
115
views
Existence of a Laplace transform that takes specific values on the integers
The classical Marcinkiewicz theorem (1939) states that if a random variable $X$ has a Laplace transform/characteristic function of the form $\mathbb{E}(e^{tX})=e^{P(t)} $ with $P$ a polynomial, then ...
- 593
8
votes
1
answer
1k
views
Uniqueness of a Solution for a Convex Optimization Problem
I have the following convex optimization problem:
$$\begin{array}{ll} \text{maximize}_{{f,g}} & \displaystyle\int_{\Omega} g^u{f}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\...
- 359
3
votes
3
answers
593
views
An integral involving hyperbolic functions
I am wondering if it is possible to obtain a closed-form formula for
$$
f(\alpha) = \frac{1}{{\sqrt{2 \pi } \; \alpha }} \int^\infty_{-\infty} x^2 \cosh(x) \; e^{-\frac{\sinh ^2(x)}{2 \alpha ^2}} \...
3
votes
1
answer
156
views
Measurability of a particular set generated by discrete probability measures
Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably ...
- 33
8
votes
3
answers
934
views
Question about Wasserstein metric
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.
My ...
- 345
8
votes
2
answers
2k
views
Median and mean of the sample mean of i.i.d. log-normal
Let $y:=\frac1n\sum_{i=1}^n x_i$, where $\{x_i\}_{i=1}^n$ is a set of i.i.d. random variables, and every $x_i$ has a lognormal distribution $x_i \sim\text{Lognormal}(\mu,\sigma^2)$. Let $\text{Med}[y]$...
- 2,239
4
votes
0
answers
516
views
Sum of Binomial random variable CDF
Suppose there are two independent Binomial random variables
$$
X\sim Binomial(n,p)\\
Y\sim Binomial(n,p+\delta)
$$
where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$.
Now ...
- 103
1
vote
1
answer
122
views
Variance bound of a functional
$X_1,\ldots,X_n$ are i.i.d standard normal random variables.
$a_1,\ldots, a_n$ are constants with $a_i \in [\kappa_1, \kappa_2]$ for all $i$ and $\kappa_1>0$.
$\hat c_n$ is given as the solution ...
10
votes
0
answers
742
views
Torus Graph Dynamics
Consider the torus graph, or the toroidal grid, which looks like
(The graph's vertices are the bold dots).
I will discuss only square torus graphs, where there is an equal number of vertices in a "...
- 403
3
votes
0
answers
186
views
Anti-concentration for sum of t-wise independent uniform variables
Let $X_{1},\ldots,X_{n}$ be i.i.d. random variables, each variable is uniform over the set of integers $\{ 0,\ldots,D-1 \}$. Let $S = \sum_{i=1}^{n}X_{i}$.
By ``small ball probability'', we have that ...
- 225
2
votes
1
answer
322
views
Questions about Levy measure in the canonical representation of infinitely divisible distributions
Let $X$ be a random variable with infinitely divisible and symmetric distribution $F$ distributed on $\mathbb{R}$.
It is well known that the characteristic function of $X$ has a canonical ...
- 671
3
votes
1
answer
196
views
Minimizer of two random walks
Consider the following two random walks:
The first random walk $\{S_n\}$ has i.i.d. step size
$$
X_i\sim\mathcal{N}(1,1)
$$
The second random walk $\{S'_n\}$ has i.i.d. step size
$$
Y_i\sim\mathcal{...
- 103
12
votes
2
answers
1k
views
Should you bet in poker against Darth Vader?
This is a theoretical question about poker-type games. I'm not going to specify the rules. You can consider No Limit Texas Hold'em or some simple theoretical model, where each player holds a number ...
- 18.9k
10
votes
2
answers
544
views
Does the optimal strategy converge in poker if the SPR tends to infinity?
This a a theoretical question about poker type games.
I'm sure I don't have to explain the rules - you can consider No Limit Texas Hold'em or some simple theoretical model, where each player holds a ...
- 18.9k
2
votes
1
answer
302
views
Order of independent random variables
Let $(p_\pi)_{\pi\in S_3}$ be given nonnegative reals such that $\sum_{\pi \in S_3} p_\pi = 1$. What are necessary and sufficient conditions for there to exist independent random variables $X_1,X_2,...
- 9,720
6
votes
1
answer
2k
views
Minimizing KL divergence: the asymmetry, when will the solution be the same?
The KL divergence between two distribution $p$ and $q$ is defined as
$$
D( q \| p)\int q(x)\log \frac{q(x)}{p(x)} dx
$$
and is known to be asymmetry: $D(q\|p)\neq D(p\|q)$.
If we fix $p$ and try to ...
- 185
2
votes
1
answer
153
views
A problem about normal distribution, independent random variables
Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Is it true that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $...
- 175
1
vote
0
answers
265
views
Time-inhomogeneous and state dependent Markov chain
We look at an inhomogeneous Markov chain $X_{n}$ that evolves according to the following transition probabilities:
$$
P(X_{n+1}=k+1|X_{n}=k)=\frac{f(k)}{n+1}\\
P(X_{n+1}=k|X_{n}=k)=\frac{n-f(k)}{n+1}\\...
- 11
5
votes
0
answers
204
views
anti-concentration of multi-linear polynomials in Gaussian variables
A Gaussian variable $X_i\sim {\cal N}(0,1)$ is anti-concentrated in the following sense: for any $\epsilon>0$ we have:
$$
\mathbf{P}( |X_i| \leq \epsilon ) = O(\epsilon).
$$
Hence if we consider a ...
- 445
2
votes
1
answer
114
views
Does maximizing $D_u$ imply stochastic ordering?
Let $\mathscr P _0$ and $\mathscr P _1$ be two non-overlapping sets of probability distributions defined on $(\Omega,\mathcal{A})$. Consider the distance defined as $$D_u(P_0,P_1)=\int_\Omega \left(\...
- 359
1
vote
1
answer
234
views
When is the second largest Gaussian r.v. the largest in the stochastic sense?
Let $X_1, \ldots, X_n$ be jointly Gaussian, each of which is marginally distributed as a standard Gaussian $N(0,1)$. It is well known that $\max |X_i|$ achieves the maximum in the stochastic sense if $...
- 773
0
votes
1
answer
165
views
Efficiently Sampling of Multivariate Distributions in the Vicinity of a Manifold
I am given a multivariate distribution, that maps each point of $\mathbb{R}^n$ to its probability of being drawn as sample and, the convolution $\mathcal{S_r}\times\mathcal{M}$ of a manifold $\mathcal{...
- 13.2k
2
votes
1
answer
216
views
Measure space for trees and other algebraic datatypes
Given a measure space $\mathcal M$, I am wondering what kind of measure space $\mathcal T(\mathcal M)$ one could associate to the set of binary trees with elements from $\mathcal M$ at each node.
The ...
- 1,241
2
votes
1
answer
369
views
Recovering a distribution from sample averages?
I'm working on a problem where I have $n^2$ real numbers $x_{11},...,x_{nn}$, all drawn i.i.d. from the same distribution $F$. I don't observe each $x_{ij}$, but I do observe the $n$ means:
$$\bar{x}...
- 371
6
votes
0
answers
133
views
Random Balanced Assignment
A balanced assignment from from $N$ objects to $K$ classes is a mapping $\sigma\colon \{ 1, \ldots, N\} \rightarrow \{ 1, \ldots, K\}$ such that
$$
\textrm{Card}( \sigma^{-1} \{j \} ) = \textrm{Card} ...
- 1,127
15
votes
2
answers
1k
views
Sum of independent random variables
We know that the sum of two independent normal random variables is again a normal random variable. But is the reverse right? If $X$ and $Y$ are independent random variables satisfying $X+Y$~$N(\mu,\...
- 153
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
1
vote
1
answer
376
views
Invariants on the space of probability distributions
There are many valuable functionals on the space of probability distributions which are invariant under variable transformations. (as an example KL divergence)
But all these functionals are defined ...
- 41
1
vote
1
answer
183
views
Diffuse measure space as a product of $[0;1]$ and another diffuse measure space
The title speaks of itself. How far is an arbitrary finite diffuse measure space from being almost isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable ...
- 1,959
1
vote
1
answer
314
views
Solution of bimodal and multimodal Weibull distribution
Is there any closed form solution for $\sigma$ in a bimodal Weibull distribution function written in the following form:
$$ P(\sigma) = 1- exp\Bigg(-\alpha\Big(\frac{\sigma}{\sigma_1}\Big)^{m1} -\...
- 33
3
votes
0
answers
79
views
Finding analytic expressions for the cumulants of a correlated random variable
I am working with cumulants of a distribution. I have an example of how the second cumulant may be simplified from:
$k_2 = p\alpha^2\left\{\left(\sum a_i\right)^2 - 2\sum_{i<j}a_ia_j\left(1-\rho^{...
- 139
4
votes
2
answers
258
views
What theorem can be used to explain this occurrence?
I'm not highly versed in research-level mathematics. I do conduct research in cellular biology. I was wondering if you could help me find a term that can be referred to when discussing the following ...
- 49
0
votes
0
answers
65
views
Wanted: example of a non-stationary sequence with reverse empirical measure
Assume we have a sequence $\xi=(\xi_1,\xi_2,\dots)$ of random variables such that $$\eta=\left(\frac{\sum_{i=1}^n \delta_{\xi_i}}{n}\right)_{n\geq 1}$$ is a reverse-martingale with respect to its own ...
- 211
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
- 1,124
2
votes
1
answer
652
views
Concentration inequality for subgaussian^4
Let $X_1,...,X_N$ be IID, mean-zero random variables whose tail is bounded by a subgaussian-tailed variable to the fourth moment, i.e., for some $t \ge t_0 > 0$
$$
P(|X_i| > t) \le C\exp\left( -...
- 719