Questions tagged [uniform]

The uniform distribution describes a random variable that is equally likely to take any value in its sample space.

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Is there a continuous function that accepts a single uniform random variable and returns two independent uniform random variables?

I can define a function $f(X) = (Y_1,Y_2)$ that accepts a random variable $X$ with a uniform distribution on $[0,1]$, and returns two independent uniform random variables $Y_1,Y_2$. This function ...
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When should I use the Normal distribution or the Uniform distribution when using Xavier initialization?

Xavier initialization seems to be used quite widely now to initialize connection weights in neural networks, especially deep ones (see What are good initial weights in a neural network?). The ...
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286 views

Copulas for generating uniform random variables with correlations

I want to generate uniform random variables which have a correlation structure defined by a graph i.e. a variable is only correlated with its neighbors in the graph and is uncorrelated with the rest ...
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416 views

Generating random matrices with specific equality constraints

Suppose I want to generate a nonnegative $n \times n$ matrix $\mathbf A$ for an odd $n$ (say, $n=5$ for a good enough example), such that the individual elements are drawn from a uniform distribution ...
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Derivation of Olsens LS Selectivity Correction

There are many estimation procedures that correct for sample selection. The most famous is Heckman's two-step selectivity correction (in two equations) that assumes bivariate normality of the error ...
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125 views

Are these equivalent (for p-values): super-uniform, stochastically larger than / dominating the uniform, conservative?

In the literature and online, I have found three different wordings that I think refer to the same concept: stochastically larger than uniform (which I take is ...
4
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35 views

PDF for the ith ORDERED uniformly random sample compared to an evenly spaced sample

Let $r_1 ≤ r_2 ≤ ... ≤ r_N$ denote an ORDERED set of N realizations of real numbers that are uniformly random on the number line from 0 to 1. Let $R_1 < R_2 < ... < R_N$ denote a set of ...
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2k views

Generate Beta distribution from Uniform random variables

I need to generate random numbers from Beta distribution using random variables from Uniform distribution. If I have two random variables $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$, and If $Y_1+Y_2&...
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How to measure the asymmetry of data distribution in a convex?

I have some 2d points data and I generated a convex hull mesh. Looking by eye, it seems that the points are not uniformly distributed inside the convex. I wonder what is the best way to characterize ...
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98 views

Markov Chain Monte Carlo (MCMC): How many samples are needed to get a uniform sample?

I am interested in a general answer although my question is rooted in a specific document. I am using the R package "hitandrun": https://cran.r-project.org/web/packages/hitandrun/hitandrun.pdf On ...
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146 views

What is the probability that the $k$th element falls in a specific interval?

The question I'm referring to comes from Stack Overflow: https://stackoverflow.com/questions/8723652/estimating-number-of-results-in-google-app-engine-query In short: With $N$ ordered samples of a ...
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34 views

continuos uniform distribution pdf value at upper bound

What is the most formal (and coerent with probability theory) definition for the value of pdf(b) where b is the upper bound of the support of the continuos uniform distribution U(a,b) ? We can choice: ...
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304 views

Distribution of the $L^2$ norm of a vector of components drawn from uniform distributions

We consider a random vector $\vec{v} = \left(x_{1}, x_{2}, \dots, x_{n}\right)$ built from $n$ real random variables drawn from a real continuous uniform distribution $\mathcal{U\left(a, b\right)}$, $...
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296 views

Does $\cos(U)$ have the same distribution as $\sin(U)$, when $U \in (0, 2\pi)$?

Consider an uniformly distributed variable $U$ in $(0,2\pi)$. My impression is that $\cos(U)$ have the same distribution as $\sin(U)$. Is my assumption correct?
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239 views

Expectation of roots of a quadratic equation

The quadratic equation $x^2 -ax+ b = 0$ is known to have two real roots, $X_1$ and $X_2$ $(X_1 > X_2)$ but the coefficient $b$ is a positive unknown and can be assumed to have a uniform ...
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155 views

Distribution of X-U(0,1) conditioned on sigma algebra of Y/X, where is Y is U(0,1)?

The question I have is: Define X,Y to be two independent uniform(0,1) random variables and $Z:=\frac{Y}{X}$ Compute $P(X<x|\sigma(Z))$ The answer given apparently by "straightforward elementary ...
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162 views

Check that a statistic is complete

I have a question regarding completeness of a statistic. So the problem is: $n$ numbers are chosen randomly and independently between $a$ and $b$ ($0 < a < b$) but the information about $a$ and ...
3
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1answer
248 views

Simplest possible (uniform) sampling of the interval [0,1] with n points

The simplest possible sampling of a function in the region [0,1] for the purpose of finding the area under the curve, for instance, could be just take the left offsets of the bars. That is, you have ...
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114 views

Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?

I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by $$ \pi_G(\xi|\mu,\sigma) = {1\...
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How to derive an estimator for the parameter of a continuous uniform distribution

$X_1, X_2,\dots.,X_n$ are i.i.d. random variates drawn from a continuous uniform distribution over $[0,\theta].$ The sufficient statistic is denoted $\max$. I want an estimator $e$ of $\theta$ that ...
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German tank variant: estimate resolution of camera given cropped photo sizes

Make whatever assumptions you like, but I like the flavor of nonparametric techniques. I have a list of the $x_i$ by $y_i$ resolutions of a number of photos, all cropped from photos taken at the same ...
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Where is the uniform distribution with one parameter ($U(\theta, k \theta)$) useful for modelling?

I recently came across the distribution $U(\theta, k \theta)$ (where k is known) in the context of statistical theory (as a nice toy example for finding MLE and the likes). However, I was wondering ...
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2k views

Finding Expected Value of a discrete uniform random variable

I am stuck on a problem for my Statistical theory class. So the problem goes like this: Let X be the discrete uniform random variable, namely, X has the pmf: $f(x)=\frac{1}{\theta}, x=1,2,...,\...
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1answer
4k views

Find posterior distribution for uniform distribution

Given X with uniform distribution in the interval [μ,μ+θ]. Suppose θ is given. Find the posterior distribution with prior distribution on your own. From that, find the Bayesian estimator with ...
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Theoretical justification behind assuming that the data is locally uniformly distributed, as seem to be used by manifold learning community

In at least three or more papers I've been studying that introduced novel algorithms for the estimation of intrinsic dimensionality (ID) based on nearest neighborhood (NN) techniques, I observed that ...
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81 views

The joint distribution of two different sums of the same independent uniform random variables

Given $k$ independent, uniform random variables $X_i \sim U(-a,a)$, $i=0,\dots k$, and two sets of coefficients $\{\alpha_i\}$ and $\{ \beta_i\}$, let $U = \sum_{i=1}^k \alpha_i X_i$ and $V = \sum_{i=...
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51 views

intuitive explanation for expected value of the square of a uniform variable

I'm confused about something that should be simple. Suppose I have a random uniform variable $X$ on $[0,1]$. It's fairly clear that the expected value of $X$ is 1/2. By integrating $x^2$ on $[0,1]$, I ...
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67 views

Transformation of probability distribution

I have a question about a snippet on page 526 in the PRML book of Bishop. Can someone explain to me why the right-hand side of equation (11.6) equals $z$? It's unclear to me where this derivation ...
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33 views

The probability of photon collision

I was reading a textbook and I couldn't figure out something that seemed really obvious: Assume that the space is uniformly and randomly filled with stars, and the mean radial distance between the ...
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107 views

Sufficiency and completeness of distribution

Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf $p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\ ...
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Proof help: Coincidences in higher dimensions

Background I recently watched a 2014 Talk by Geoffrey Hinton (a key researcher in Machine Learning literature) where he discusses the concepts behind the recently published Capsule Networks. In the ...
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1answer
81 views

It is true that $\mathbf X \sim F_X \Rightarrow F_X(\mathbf X) \sim U_{[0;1]}$; does the converse hold for multivariate $\mathbf X$?

For a univariate real-valued random variable I am pretty sure that the converse holds. Consider a multivariate $\mathbf X$ with values in $\mathbb R^n$ with measure $\mu(X)$ and its multivariate CDF ...
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472 views

Approximating Uniform Distribution with Mixture of Gaussians

Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$. Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$ \...
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1k views

Finding UMPT for uniform distribution with varying support

$\textbf{Problem}$ Let $X_1,\dots,X_n$ be a random sample from $f(x;\theta) = 1 / \theta$, where $0 < x < \theta$. We want to test $H_0: \theta \leq \theta_0$ versus $H_1: \theta > \theta_0$. ...
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264 views

How to check the correctness of calculations with a gamma distribution?

I’m reading Ponomareva, Roman, and Date (2015) and trying to generate vector $P$ of the $2Ns + 3$ probability weights: $$P =\{\underbrace{p_1, p_2, \ldots, p_s, p_1, p_2, \ldots, p_s,p_1, p_2, \ldots, ...
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85 views

Inconsistent results with Monte Carlo solutions to similar problems in probability

I am presently going through the book Fifty Challenging Problems in Probability with Solutions and implementing Monte Carlo solutions to most of the problems in R to get familiar with the language, ...
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285 views

How to find a conditional CDF of a trapezoidal distribution?

I'm working on a game theory model of imperfect information, where players observe certain attributes via noisy signals. Specifically, Player 1 has the opportunity to choose any value $\eta$ from the ...
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541 views

Empirical multivariate probability integral transform

Is there a 'simple' way to obtain a non-parametric empirical multivariate probability integral transform? Univariate case The probability integral transform relates to the transform of any random ...
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0answers
535 views

Finding MLE with ordered statistics?

Let Y1 < Y2 < ... < Yn be the order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval: $$[\theta - \rho, \theta + \rho]$$ ...
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399 views

Distribution of correlation coefficients for uniform random variables

Let $n>1$, let $X$ be uniformly distributed on $[-\frac12,\frac12]$, and consider the sequence $X_1,\ldots,X_{n+1}$ of independent copies of $X$. R implements ...
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72 views

Transforming a uniform-on-sphere random vector

Consider the 3-D real random vector $(X_1,X_2,X_3)$ which is uniformly distributed on the surface of a unit sphere. What can be told about the distribution of $(aX_1,bX_2,cX_3)$, where $a,b,c,$ are ...
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Interarrival time distribution of uniform arrival process

I am currently trying to model an uniform arrival process within my simulation model. However, I can only model it by means of an interarrival time (I can let the model wait for a certain amount ...
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168 views

Obtain order statistics using uniform order statistics

This is a homework questions. Can you guys give me some hints? Let $U_{(1)}<\cdots<U_{(n)}$ be the order statistics of a sample of size $n$ from a Uniform$(0,1)$ population. Show that $F^{-1}(...
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162 views

Should percentiles of one set of samples from a distribution wrt another set be uniformly distributed?

I'd like to pose the following question which for some reason is proving to be unclear to me. Assume we have the Normal distribution, mean 0, sd 1. Let's say we take 1000 samples from it; call them $...
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124 views

Multi-information of a uniformly distributed random variable on the L1 sphere

I posted this question in the stackexchange mathematics forum without any reponse. Maybe it was the wrong forum, so I try it here. I tried to compute the multi-information (MI) $I[\mathbf U] = \sum_{...
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34 views

Compute the two following summation with uniform and normal random variable

I've to show, if possible, that these two statements hold: Show that if $x \sim Uniform(0,1)$ is a data sample of size 10x10, further vectorized, then $$100 \le \int p_z(g)\sum_{i=1}^{100} (2x_i^2 + ...
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Why does deviation from uniform distribution suggest skewed-t model may not provide adequate fits for copula model

I read a book titled "Statistics and Data Analysis for Financial Engineering with R examples". At page 203, I read the following paragraph. "Figure 8....
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Properties of the diff of a sorted uniformly generated set

I am studying a set of uniformly generated points, more concretely the distance between the points. When the set is unsorted the histrogram shows it is normally distributed and that matches my ...
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86 views

Convergence in Probability (Analytical Solution Verification)

Problem: Let $X_1,X_2,\cdots$ be independent random variables that are uniformly distributed over $[-1,1]$. Show that the sequence $Y_1,Y_2,\cdots$ converges in probability to some limit, and identify ...
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Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...