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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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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Is there a continuous version of the Uniform distribution?

The Uniform distribution is not differentiable. Is there a differentiable distribution that approximates a Uniform?
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Does uniform conditional distribution imply independence?

Take two random variable $X,Y$ and suppose $X$ is distributed uniformly on $[0,1]$ conditional on $Y$. Does this imply that $X$ is independent of $Y$? Could you make an example?
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Normal Distribution with Uniform Mean

I'm trying to understand the distribution, mean, and variance of a normal random variable, with the mean parameter having a uniform distribution. Based on my R simulations it seems that this compound ...
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Verifying that a random generator outputs a uniform distribution

I asked a student of mine this question: If you have a random number generator that outputs a number between $1$ and $k$, how would you write a test that decides whether the generated ...
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Log-uniform distributions

I am having some difficulty understanding what log uniform distributions are. Suppose that $\log X$ is uniformly distributed on the interval $[1,e]$. How do I describe $P(X=x)$? It seems like there ...
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How to efficiently choose $n$ subset out of a set of $m$ many numbers, in a randomized uniform manner?

Problems: It is fairly simple: we have a list of numbers $x_1, x_2, \ldots,x_n,\ldots, x_m$. Our goal is to randomly and uniformly choose a subset of $n$ many numbers out of these. This means that, ...
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Can I use Kolmogorov Smirnov test to check if my data are uniformly distributed?

I'd like to check if distribution of my data is significantly different from a uniform distribution. I know that the K-S test is used for checking the normality of data, but I wonder if it can be ...
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If ϵ is uniformly distributed, then a linear probability model is appropriate? Can I find any Literature?

A latent variable model involving a binomial observed variable $Y$ can be constructed such that $Y$ is related to the latent variable $Y^*$ via $ Y = \begin{cases} 0, & \mbox{if }Y^*...
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Derivation of Rayleigh-distributed random variable

I only have a uniform distribution function between [0,1]. And from this distribution, I should generate a sequence of Rayleigh distributed random variable using some software. Anyhow, I was able to ...
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Computing variance of squared difference of i.i.d. uniform random variables

When $U$ and $U'$ are two i.i.d. uniformly distributed random variables on $[0, 1]$, show that $$\mbox{Var} \left( (U-U')^2 \right) = 0.04 $$ I tried plugging in the formula $\mbox{Var}(U^2)=E(U^4)-E(...
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Probability density of compound triangular distribution with uniformly distributed mode?

What are the probability density function and cumulative distribution function of a compound triangular distribution with uniformly distributed mode, both supported on $(-a, a)$? I.e., $$ m \sim \...
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Finding pdf of transformed variable for uniform distribution

This is from MITx's Intro to Probability and Statistics course, the problem is on this page. Suppose $X \sim \textrm{Uniform}(0,1)$ and $Y=X^3$. Find the pdf for $Y$. Since it's a uniform ...
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Probability of uniformly drawing N numbers less than the expected second highest value

In the case of 3 draws (N=3) from Uniform[0,1], the expected second highest value would be 1/2. Although unlikely it could happen that all three numbers were less than 1/2. It is exactly this ...
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Limits of integration of a density function

My question is based on this post. In summary, $X \sim \text{Unif}(a,b)$ and $Y|X \sim \text{Unif}(a,X)$. Then the author does the following calculations: \begin{align} f(y) = \int_{-\infty}^{\infty} ...
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Improving Chebyshev-type bound for discrete uniform distribution

I take $N$ samples from a fully specified, discrete, finite uniform random variable $X$ with mean $\mu$ and variance $\sigma_X^2$. I want to find the probability that the absolute error of the ...
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Expected value of $Ye^X$ where $X \sim U(0,1)$ and $Y \sim U(0,1)$

I am trying to find the expected value of $Z$ where $Z = Y\cdot e^X$ where $Y \sim U(0,1)$ and $X \sim U(0,1)$. My attempt so far: $$F_Z(z) = P(Ye^X \le z) = \int \int_{Ye^X \le z} f(x,y)\, dxdy$$ ...
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Testing whether sampling (convex polytope) is uniform

Currently, I am sampling points from: i) a convex polytope (i.e. Ax <= b) ii) a high dimensional simplex The algorithms I am using are hit-and-run and a simple version of Bayesian bootstrap. I ...
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Asymptotic distribution of uniform order statistics

It can be shown that for an iid sample from a Uniform(0, 1) distribution, \begin{equation} n(1-U_{(n)}) \rightarrow exp(1) \\ n(U_{(1)}) \rightarrow exp(1) \end{equation} To see this just try finding ...
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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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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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How to perform goodness of fit test and how to assign probability with uniform distribution? [duplicate]

I have to demonstrate that a generator of VoIP calls generates calls uniformly distributed between callers. In particular the distribution is the uniform (min, max) one where the volume per caller ...
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Difference between Excel's RAND(), RAND()*RAND(), etc

I plotted below the standarized results of: RAND() RAND() * RAND() ... RAND() * RAND() * RAND() * RAND() * RAND() * RAND() It seems that the results are getting to zero, is that because you're ...
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Probability that $2\times2$ matrix of random variables is invertible

Let $X_1, X_2, X_3, X_4$ be random variables, and let $A$ be the following matrix: $$ \left[\begin{matrix} X_1 & X_2\\ X_3 & X_4 \end{matrix}\right]. $$ Assume that $X_1, X_2, X_3, X_4$ are ...
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What probability distribution is to the discrete uniform distribution as the beta distribution is to uniform distribution over $[0,1]$?

A beta distribution with its parameters $\alpha = \beta = 1$ is the uniform $[0, 1]$ distribution. What distribution is to the discrete uniform distribution (the sample space is left undecided), as ...
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PDF of cosine of a uniform random variable

There is a formula for the density of the cosine of random variable that's a uniform on $(-\pi,\pi)$ as discussed in this page: $f_{Y}(y) = \dfrac{1}{\pi \sin(\cos^{-1}y)}, y \in\ [-1,1]$ Can anyone ...
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Can the difference of random variables be uniform distributed? [duplicate]

Given two random variables X and Y with some distribution D, is it possible to choose a D such that Z = X - Y is uniform? Is there a standard distribution D that would satisfy this?
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How to measure whether a discrete distribution is uniform or not?

Say I have two vectors [1,2,1,2,2] and [1,2,1,1,1]. The number at each dimension is the frequency of one element. How do I measure whether these two vectors are close to the uniform distribution? I ...
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Generating uniform points inside an $m$-dimensional ball

The present question follows on from some other questions on this site asking how to generate uniform points on a disc (see e.g., here, here and here). The natural extension of that problem is to ...
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Confidence interval for mean of a uniform distribution

I've been trying to compute a 95% confidence interval for the mean of a height sample, which is uniformly distributed. I have calculated the following sample statistics: $$n=10 \quad \quad \bar{x} = ...
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Uniform random variable distribution

This is a homework problem out of the book. It says If $U$ is a uniform random variable on [0,1], what is the distribution of the random variable $X = [nU]$, where [$t$] denotes the greatest ...
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Asymtotic distribution of the MLE of a Uniform

A property of the Maximum Likelihood Estimator is, that it asymptotically follows a normal distribution if the solution is unique. In case of a continuous Uniform distribution the Maximum Likelihood ...
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Continuous uniform random variables convergence question

Let $X_1, X_2, \ldots$ be independent $U(0,2)$ random variables and let $$Y_n = \prod_{i=1}^n \, X_i \;.$$ How do I prove or disprove that that $Y_n$ converges to $0$ almost surely ?
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How to compute the distribution of a function of multiple random variables?

$X$ and $Y \sim U(0,1)$. Let $$\eqalign{ g(x,y) &= x &\text{ if } &x^2+y^2 \le 1 \\ &=2 &\text{ if } &x^2+y^2 \gt 1 }$$ and $Z = g(X,Y)$. How to find $F_Z(z), \...
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MLE of $f(x\vert\theta)=1/\theta$, $x_1 , \cdots , x_n \sim U(0,\theta) \;\;, \theta>0$, [closed]

Original question $x_1 , \cdots , x_n$ are independent random variables, identically distributed as a uniform distribution over $(0,\theta)$. $$ f(x \vert \theta) = \frac{1}{\theta}, \; 0<x<\...
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X is Uniform $[-\theta,\theta]$ what is the distribution of $Y=\frac{1}{x^{2}}$?

X is Uniform $[-\theta,\theta], \theta>0$ what is the distribution of $Y=\frac{1}{x^{2}}$ So I've been working on some transformation questions; however, most of them have been one to one. I am a ...
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Convergence to a Uniform Distribution

$\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} $ Show that if $P(X_n = i/n)=1/n$ for every $i = 1,...,n$, then $X_n$ converges in distribution to a uniformly distributed random variable $X$. ...
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Sufficient statistics for Uniform $(-\theta,\theta)$

So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ? In other ...
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Apparent inconsistency arising from showing that $x_{(n)}$ is sufficient for $\theta$ where $X \sim \frac{1}{\theta}\mathbb{I}_{(0, \theta)}$

The problem is to show that the largest order statistic $x_{(n)}$ is sufficient for $\theta$ where $X \sim \frac{1}{\theta}\mathbb{I}_{x \in (0, \theta)}$ is a uniform distribution. I believe I have ...
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Bivariate random vector uniform distribution

I have a question regarding the definition of a uniform distribution for a bivariate random vector. For example, I am doing a few exercises and the premise of the questions are as follows: Let $(X, ...
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Estimating Uniform distribution endpoints using data with errors

Suppose I have a random variable $X$ ~ $Unif(0,\theta)$ where I want to estimate $\theta$. I draw a sample $X_1,...,X_n$.One way is to get a point estimate using e.g. maximum likelihood estimation ...
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Statistical test for uniform distribution

I have a sample of 5 numbers from known interval [0, 10]. Is 5 numbers is enough to make some conclusions about whether these numbers are drawn from uniform distribution or not?
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Last-digit test and probability

I have done a last-digit test on a draw of numbers between 1 og 370. Team A drew 64 numbers from the pool, and the last digit of those numbers spread out like this: Digit: 0 1 2 3 4 5 6 7 8 9 ...
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Two dependent uniformly distributed continuous variables and Bayes' theorem: a billiard table exercise

I am trying to solve the following exercise from Judea Pearl's Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. 2.2. A billiard table has unit length, measured from ...
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How does the inverse transform method work in discrete r.v.?

In this question How does the inverse transform method work? it's mentioned the general procedure to generate r.v. U <- runif(1e6) X <- qnorm(U) X How ...
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Limiting distribution of a ratio using Basu's theorem

Edit: there's seems to be a typo in original question. This is a past exam question that I'm trying to solve. Suppose that $X_1,\ldots, X_n$ are i.i.d. Uniform (0, $\theta$) random variables. Let $...
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Uniform distribution with Gaussian Priors

Let's say i've got a uniform distribution defined as follows $$X \sim U[\min (\theta_1,\theta_2),\max (\theta_1,\theta_2)]$$ I've also got that $\theta_1,\theta_2$ are i.i.d zero-mean normal ...
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Finding complete sufficient statistic

Let $X_1 , ....,X_n$ be iid. $Uniform[-\theta,\theta]$. I need to find the complete sufficient statistic for $\theta$ or prove there does not exist such. I know that $T=(X_{(1)}, X_{(n)} )$ is a ...
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On the generality of using empirical FDR even for conservative p-value distributions

Given the discussion here: Why are p-values uniformly distributed under the null hypothesis? And, particularly the point that @whuber brought up on the composite hypothesis (link here). I'm ...
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Do the data from these two different samplings follow the same distribution?

I have two approaches for data sampling: Sampling from a uniform distribution in $[0, 1]$ and rejecting values outside a certain limit, i.e. $0.50<p<0.51$. Sampling from a uniform distribution ...

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