All Questions
Tagged with pr.probability probability-distributions
1,384 questions
1
vote
0
answers
146
views
Using maximum entropy principle for joint probability estimation
Let $X_1, \dots, X_n, Y$ be random variables, each taking values in $\{0,1\}$. Assume that we are interested in estimating, for each $v=(v_1,\dots,v_n)\in \{0,1\}^n$, the probability
$$
p(v) = P[Y=1|...
- 7,182
1
vote
1
answer
157
views
Moments of rescaled Bernoulli random matrix
Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z_{ij} \sim \text{Bernoulli}(p) \implies X_{ij} = \frac{Z_{ij}}{\sum_{k=1}^m Z_{ik}}.$$
Is ...
- 269
1
vote
2
answers
277
views
Distribution of interarrival times for a special class of stochastic point processes
I am interested in Poisson-binomial stationary point processes (here on the real line) defined as follows. Let
$t_k=k/\lambda$, with $k\in\mathbb{Z}$ and $\lambda>0$,
$F_s(x)$ be a symmetric, ...
- 3,098
5
votes
1
answer
392
views
Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
- 1,334
6
votes
3
answers
447
views
Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace
Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
- 6,853
0
votes
1
answer
133
views
How to demonstrate a correlation inequality? [closed]
If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$.
The correlation between Z, Y is greater than between X, ...
- 11
3
votes
2
answers
593
views
A lower bound for the expectation of $\min\{X,n-X\}$ when $X$ follows a $\mathrm{Binomial}(n,p)$ distribution
Let $X$ be a random variable following a $\mathrm{Binomial}(n,p)$ distribution, and let $$Y=\min\{X,n-X\}.$$ Ispired by the problem posed by C. Clement on https://math.stackexchange.com/questions/...
- 377
3
votes
2
answers
297
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Does my construction always result in a stationary Poisson point process of intensity $1$? How so?
My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
- 3,098
6
votes
1
answer
261
views
Convergence speed of the tail of distribution using Tauberian remainder theorem
This question may be related to this one.
Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem.
Let $f$ be some one-sided probability ...
- 284
2
votes
1
answer
89
views
Probability measure of trapezoidal area [closed]
Let $Pr_{(X,Y)}$ be a probability distribution of a random vector $(X,Y)$. Let $F$ be the cumulative distribution function of $(X,Y)$. Define
$$
\mathcal{A}\equiv \{(x,y): x\leq 2 \text{ and }x-y\leq ...
- 108
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\...
- 161
1
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0
answers
100
views
Exponential decay of a random matrix falling into a ball
Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that ...
- 1,495
7
votes
1
answer
347
views
Expectation for game choosing uniformly number in $[0,1]$ until it decreases
We are playing a game where we keep on choosing a number from the uniform distribution U(0,1). The game goes on until we have the current number less than the previously picked number, i.e. the game ...
1
vote
1
answer
252
views
Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible
$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...
- 151
2
votes
0
answers
192
views
Convergence of Gibbs distribution to Dirac measure [closed]
Consider the probability density function on $R^d$ for a continuous function $F: R^d \to R$:
$$
q_{\varepsilon}(x) = \frac{1}{Z} \exp\left(-\frac{1}{\varepsilon} F(x)\right).
$$
Denote $x^* = \arg \...
- 31
1
vote
1
answer
158
views
Generating iid random vectors such that the distribution of their dot product is $\mathit{Uniform}[a, b]$
Take two independent and identically distributed random vectors $X_i$, $X_j$.
I want to find a multivariate distribution for these vectors such that the dot product $X_i^\top X_j \sim U[a, b]$.
This ...
- 119
1
vote
1
answer
423
views
Generalized random harmonic series
Let $Z_n=\sum_{k=1}^n a_k X_k$ with $(a_k)$ a strictly decreasing sequence of positive real numbers that tend to zero. The random variables $X_k$ are independent and satisfy $P(X_k=1) =p_k, P(X_k=-1)=...
- 3,098
1
vote
1
answer
278
views
Construct a random vector as a function of another random vector
ASSUMPTION 1: there exists a continuous random vector $(X,Y,Z)$ such that
$$
\begin{cases}
p_1=\Pr(X\geq 0, Z\geq 0)\\
p_2=\Pr(Y\geq 0, Z< 0)\\
p_3=\Pr(X< 0, Y<0)\\
\end{cases}
$$
where $(p_1,...
user447159
3
votes
1
answer
526
views
Wasserstein-type concentration inequalities for empirical measures on polish spaces
Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...
- 5,405
4
votes
2
answers
218
views
Do these distributions have a name already?
In playing with some math finance stuff I ran into the following distribution and I was curious if someone had a name for it or has studied it or worked with it already.
To start, let $\Delta^n$ be ...
- 646
1
vote
1
answer
101
views
Estimating the average of two gaussians' mean with minimal squared error
This is a follow-up to my previous question.
Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
- 618
0
votes
1
answer
255
views
Sufficient conditions for decomposition of a bounded random variable into several small pieces
Given a random variable $X$ with $\mathsf{supp}\, X \subseteq [0,1]$ and $n$ positive numbers $h_1,\cdots,h_n$ with $\sum_{i=1}^n h_i=1$, I want to know some sufficient conditions for decomposing $X$ ...
- 550
2
votes
1
answer
872
views
Estimating the average of two gaussians' mean
Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$.
In my setting, $\sigma_1,\sigma_2$ are known ...
- 618
2
votes
1
answer
1k
views
Components of a Gram matrix and its eigenvalues
The Gram Matrix is defined as $$\sum_{i=1}^n X_iX_i^T,$$ where $X_i$ is drawn from the unit sphere based according to some continuous distribution (Relation between eigenvalues and the gram matrix for ...
- 215
6
votes
3
answers
368
views
Curvature function as a random variable with uniform distribution
Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produces a probability density function
$f_g:\mathbb{R}\to \...
- 356
0
votes
1
answer
46
views
PDF of the summation of L lognormal RVs
Given the following summation
$$\gamma = \sum_{l=1}^{L} y_{l},$$
where the PDF of $Y$ follows the lognormal distribution and is given by
$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{...
0
votes
0
answers
100
views
Lognormal PDF in terms of the Meijer-G function
Is it possible to write this lognormal PDF in terms of the Meijer-G function?
$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{(-10\log_{10}(y) - \mu)^2}{2 \sigma^2}\right)$$
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
- 128k
1
vote
1
answer
159
views
A problem related to bivariate normal stochastic order
Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray*}
\boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\...
- 211
1
vote
1
answer
84
views
Jeffreys' priors as coefficients of a linear estimator
I asked the following question in a forum more suitable for statistics, but I didn't get any answer; I hope, someone could shed light on my question:
I have three random variables, $X_1$, $X_2$, and $...
user386878
1
vote
1
answer
202
views
A problem related to stochastic ordering
Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray*}
\boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\...
- 211
2
votes
1
answer
287
views
Uniform distribution on a manifold
To generate a uniform distribution on a sphere $S^n$ in $\mathbb R^{n+1}$, we can normalize a vector whose entries are $n+1$ i.i.d normal random variables. If $\rho$ is a correlation, $|\rho|<1$, ...
- 1,608
5
votes
0
answers
239
views
Expected value of $X^{\top}(XAX^{\top})^{-1}X$ for large random $X$
Let $X\in \mathbb{R}^{m\times n}$ be a random matrix where the entries are i.i.d. standard normal, and let $A\in \mathbb{R}^{n\times n}$ be a deterministic diagonal matrix with positive entries on the ...
- 161
1
vote
1
answer
123
views
Stochastic ordering of absolute multivariate normal random variables
Let $X\sim\mathcal{N}(\boldsymbol{\mu}_1,\mathrm{\Sigma}_1)$ and $Y\sim\mathcal{N}(\boldsymbol{\mu}_2,\mathrm{\Sigma}_2)$. Then it is know that $\mathbb{P}(X>\boldsymbol{t})\leq\mathbb{P}(Y>\...
- 211
1
vote
1
answer
141
views
Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?
For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
- 13
0
votes
0
answers
86
views
Expected diameter of a random point set
General problem: For a point set $S\subset X$ in a metric space $(X,d)$, let $\text{diam}(S)=\max_{x,y\in S}d(x,y)$. Given a distribution $P$ on $X$ and $m$ i.i.d. points $x_1,\ldots,x_m\sim P$, what ...
- 19
-1
votes
1
answer
74
views
Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail
Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...
- 1
4
votes
2
answers
187
views
Expected value of a ratio of squared normal and linear combination of squared normal [closed]
Given two positive constants $c_1,c_2$ and two independent standard normal random variables $a,b$, how to calculate the following expected value
$$
\mathbb{E}\left[\frac{a^2}{c_1a^2+c_2b^2}\right]
$$
...
- 161
4
votes
0
answers
96
views
Is this conjecture about the binomial and beta distributions true?
Let $X$ follow a binomial distribution with parameters $n$ and $p$, and also fix $k$ such that $1<k<n$. Define
$$a = \mathbb{E}(X-k)^+$$
and
$$b = \mathbb{E}\log\binom{X}{(X-k)^+}$$
where the ...
1
vote
1
answer
182
views
From probability distribution in $\mathbb{R}^3$ to probability distribution in $\mathbb{R}^4$
I am working on a research paper where I need to investigate conditions for the existence of probability distributions satisfying certain characteristics. I have already asked a related question (here)...
- 108
4
votes
2
answers
343
views
Concentration of $k$-th pairwise distance of random points in a unit square
For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...
- 662
3
votes
0
answers
169
views
Probabilistic behavior of greedy point selection in the plane
Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
- 4,049
0
votes
1
answer
479
views
Probabilistic interpretation of derivative of a Dirac delta function
Consider $g : \mathbb{R}^d \mapsto \mathbb{R}$ defines some surface $\Sigma$ in $\mathbb{R}^d$. Then I can define a random variable $X_1$ with support only on $\Sigma$ by using a pdf of the form $$p_1(...
- 333
3
votes
1
answer
122
views
Best approximation of normal with $m$ atoms in Kolmogorov-Smirnov distance
Let $d_{KS}(F,G)= \sup_{x} |F(x) -G(x)|$ be the Kolmogorov-Smirnov distance between two cdfs $F$ and $G$.
Question: Let $F_m$ be a cdf of distribution with $m$ atoms and let $\Phi$ bet the ...
- 671
0
votes
2
answers
341
views
Conditions for existence of a distribution with full support
Consider a $6\times 1$ continuous random vector
$$
\eta\equiv (\eta_1,\eta_2,..., \eta_6)
$$
satisfying the following property:
$$
\underbrace{\begin{pmatrix}
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix}}_{\...
- 108
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
1
vote
0
answers
81
views
Calculating the mean squared error for an estimate of a large sum
Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the ...
1
vote
2
answers
274
views
$k^{\text{th}}$ maxima of $n$ i.i.d chi-square random variables
I am trying to study the asymptotic behavior of $k^{th}$ order statistic of $n$ i.i.d chi-square distribution. Let $X_1, \cdots , X_n$ be i.i.d $\chi^2_1$ random variables and $X_{(k:n)}$ be the $k^{...
- 399
1
vote
1
answer
417
views
Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution
It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...
user325283
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
