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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Calculating the sum of dependent uniform random variables

My question derives from Problem calculating joint and marginal distribution of two uniform distributions. So, suppose we have random variables $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as ...
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Reference books on uniform spherical distributions in multiple dimensions

QUESTION What is a citation of a book whose scope includes the uniform distribution [1] that is generalized to an $n$-ball [2]? Among other things, I'd like to read a book that include such ...
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What is the probabilty that X > 2 conditioning on Y = 2? (Homework)

another homework question here. Let 𝑌 be a binomial random variable with 10 number of trials and 0.2 probability of success. Let X be a uniformly distributed random variable over the interval [0, 3]. ...
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Are differences between uniformly distributed numbers uniformly distributed?

We roll a 6-sided die a large number of times. Calculating the difference (absolute value) between a roll and its preceding roll, are the differences expected to be uniformly distributed? To ...
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How many classmates does a freshman have?

The freshmen at East China Normal University has just received their student ID. Let the last three digits of a student ID be ABC, then A is the class he is in, whereas BC is his number in the class. ...
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What is the difference of $\Sigma$ esimation of Gaussian Copula based on known CDFs VS unknowns

Recently, I read this web page which explains the Copula package in R. A question occurred to me. Consider a data set $D_{n\times d}$ which $n$ is the number of samples and $d$ is the number of RVs. ...
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Expected min distance between N Uniform RVs

Inspired by this recent question, what is the expected value of the minimum of the pairwise distances between $N$ uniform and independent RVs (uniform in $[0,1]$)? i.e. Let $X_1,...,X_N$ these RVs, ...
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What's the distribution of the closest point from uniform samples?

Suppose you have $N$ values $x_1, \ldots, x_N$ that are uniformly sampled in $[0; 1]$. For a random $x_k$ amongst the $(x_i)_i$ (with equiprobability), what is the expected value of the distance ...
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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 ...
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Does the sum of discrete uniforms coverge to a discrete Gaussian?

Is there some analogous of the Central limit theorem for discrete uniforms and discrete normal distributions? To be more specific, let's say we have identical and independent random random variables $...
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Sufficient Statistic of Uniform $(-\theta,0)$

Let $X_1, ... , X_n$ be i.i.d random variables Uniform $(-\theta,0)$ , with $\theta > 0$ parameter \begin{align}f_{\theta}(x_1,x_2,\cdots,x_n)&=\prod_{i=1}^nf(x_i;\theta) \\&=\frac{1}{(\...
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Failed to get Uniform distribution from Gamma distribution

I read in Chapter 6 in this book that $p(K)\propto 1$ is equivalent to $e^{-K}\sim Gamma(0,0)I(0,1)$ where $K$>0 and is uniform distribution, e.g., $K \sim Uniform(0,100)$; $I(a,b)$ is the ...
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Transforming a uniform random variate to points on a circle

Sample $U \sim \text{Uniform}(0,\sqrt{2}-1)$. Accept $U$ with probability $1/(1+U^2)$ (else reject and sample again). Set $X = 2U/(1+U^2)$ and $Y = 1-UX = (1-U^2)/(1+U^2)$. With probability 1/2, ...
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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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How can I generate 2 sets of variables from different distributions with a correlation between them in r? [duplicate]

I am working in R and would like to generate 40 numbers from $\mathrm{N}(0,1)$ and another 40 from $\mathrm{Uniform}(0,2)$ with a negative correlation (for example: $r = -0.45$) between them. The ...
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Using an Autoencoder with uniformly distributed data

Setup: Dataset: 40k uniformly distributed 13-dim samples (floats between 0 and 1) AutoEncoder: (input: 13dim) - fc layer 13 dim, relu - latent layer - fc layer 13 dim, relu - (output: 13dim) Loss: ...
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Limiting Distribution of $n\left[Y_n\right]$ where $Y_n$ is the minimum of a sample of size n from Uniform$\left(0,\theta\right)$ distribution

Suppose $X_1,X_2,\dots,X_n$ is a random sample from Uniform$(0,\theta)$ for some unknown $\theta > 0$. Let $Y_n$ be the minimum of $X_1,X_2,\dots,X_n$. (a) Suppose $F_n$ is the CDF of $nY_n$. Show ...
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Finding bias of $\hat\theta=\max\{x_1,\ldots,x_k\}$ where $x_i$'s are discrete uniform

I am working through some textbook problems and came across a problem I am having difficulty with. The problem asks to give the bias of a point estimate, namely for a given set of data $X = \{x_1, x_2,...
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Hypothesis testing of sum of uniform distribution

I want to have some clarifications related to below question from Casella Berger Regarding $\phi_1(X_1),$ I understand that $\alpha$-value is 0.05 and hence we are looking for a value of $C$ for ...
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1answer
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Prove that $X_{(n)} - X_{(1)}$ is an ancillary statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be an independent and equally distributed random sample whose distribution is uniform on the interval $(\theta,\theta+1)$, $-\infty<\theta<+\infty$. Then consider ...
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Estimating parameters of multivariate regression using the maximum likelihood method with a uniform distribution of residuals?

Representation of regression in a matrix form: $$Y = XA + E,$$ where: $X$ - independent variables, $Y$ - dependent variables (observations), $E$ - errors, which have a uniform distribution, $A$ - ...
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How to check if a distribution is different from uniform distribution using the values only?

I have a discrete empirical distribution, where the values D1 = {1,2,3,7,9,15,16,17,18,19,20} occurs with some different frequencies. The frequencies are not of interest to me. I am interested in how ...
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The joint pdf of sample maximum and sample mean for uniform distribution?

Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$ ...
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Draw integers independently & uniformly at random from 1 to $N$ using fair d6?

I wish to draw integers from 1 to some specific $N$ by rolling some number of fair six-sided dice (d6). A good answer will explain why its method produces uniform and independent integers. As an ...
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104 views

What is the distribution of the number of occurrences of a given integer, in repeated samples drawn from Uniform Integer [1, N] distribution?

If I draw M repeated samples from Uniform Integer Distribution on [1,N], what will be the distribution of the number of times a given integer chosen among the integers 1,2,...,N appears in my M ...
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110 views

Test of uniform distribution using KS-test and chi square in R

I want to test if a given sample $x$ of $n = 500$ continuous observations is uniformly distributed on a given interval of $[a,b]$ ($a = min(x)$ and $b = max(x)$). Therefore I would like to compare the ...
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1answer
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Generating three random numbers from normal distribution using 3 independent uniform numbers

Each of the variables A,B,C follows a normal distribution with E(A)=E(B)=E(C)=1,V(A)=1,V(B)=2,V(C)=4. Correlation between A-B=0.2,B-C=0,AC=0.8. We are given three independent uniform (on[0,1]) random ...
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UMVUE of $\cos\theta$ when $X_i\sim U(0,\theta)$

$X\sim U(0,\theta)$. To find the umvue of $\cos\theta$ is it enough to find the umvue of theta and substitute for it. Umvue of $\theta$ being $(n+1)X_{(n)}/n$, is the answer $\cos (n+1)X_{(n)}/n$?
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What is the posterior distribution of a Bernoulli prior that gets updated with a continuous uniform signal?

I'm trying to figure out what the distribution of the posterior is after I update a Bernoulli prior with a continuous uniform signal, say: P(D=G|u)=x where D{G,I} and u is uniformly distributed on ...
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Unclear “mathematical notation” in a polynomial

Although, the Enigma here is a protocol for enhancing the privacy in blockchain; however, the question is about mathematical notation, where we want to calculate the coefficients in a polynomial. ...
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1answer
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Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem

I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following: "Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
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2answers
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Discrete uniform vs binomial distribution

Situation: a box contains N balls numbered $1,2...,N$. $N$ unknown. $n$ balls drawn using SRS with replacement and number recorded. A random variable $X$ is defined as the number recorded on $ith$ ...
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Expected outcome of a process following a uniform distribution [closed]

A gambler is playing a game of roulette. There are $37$ possible outcomes, each numbered from $1$ to $37$. The probability of rolling any outcome is the same for each outcome. One game of this ...
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Calculate the constants and the MSE from two estimators related to a uniform distribution

Consider a simple random sample $X_{1},X_{2},\ldots,X_{n}$ whose distribution is given by $X\sim U(0,\theta)$. Moreover, consider the estimators $\hat{\theta}_{1} = c_{1}\overline{X}$ and $\hat{\theta}...
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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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1answer
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Finding the uniformly most powerful test for hypothesis

Let $\mathbf{X}=(X_1,...,X_n)^T$ is a simple sample where $X$ belongs to exponential distribution family $\mathcal{P}=\{ f(x;\mu,\sigma \}, -\infty<\mu<\infty, 0<\sigma<\infty.$ Density is ...
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Relation between independence and correlation of uniform random variables

My question is fairly simple: let $X$ and $Y$ be two uncorrelated uniform random variables on $[-1,1]$. Are they independent? I was under the impression that two random, uncorrelated variables are ...
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1answer
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What efficient methods are there to assert that a population has a uniform distribution?

I stumbled upon this article on confidence intervals. The concept as a whole made sense but seemed strange to me. I concluded that given a fixed method of randomly sampling the population, a fixed ...
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Checking if a minimal sufficient statistic is complete

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[-\theta, 2\theta]$ with $\theta \in \mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete. I found ...
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Can I test for inequality in H0 using chi square test?

Let's say I want to test whether an $n$-sided dice is not too unfair. In the standard chi-square test we test the zero-hypothesis $$ H_0\colon (p_1,\dots p_n) = (1/n,\dots,1/n) ,\quad\text{i.e.,}\quad ...
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Conditional expectation of uniform random variable given order statistics

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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Deriving standard normal distribution from a statistic involving normal and uniform random variables

I tried deriving distributions of numerator and denominator separately. But found that there is no closed form. I have no clue on how to show that Z is standard normal.
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Marginal derivation from joint pdf

I have a uniform prior f(Θ) ~ U(4,10) and a uniform 'observation' model f(X|Θ) ~ U(θ-1, θ+1). Their joint pdf is f(X,Θ)=1/12 for 4 < θ < 10 and (θ-1)< x <(θ+1)  and 0 otherwise. If I ...
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3answers
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Can't understand why rejection sampling works

I want to generate sample points $\{z_i\}$ in an arbitrary 2D shape, e.g. a circle centered at the origin with radius 1. Rejection sampling says: Look at 2 uniform random variables over $[0,1]$, $X$ ...
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1answer
85 views

How to generate a conditional random variable in R? [closed]

Suppose there is a sample $X\sim N(0,1)$ x<-rnorm(100). If I want to generate a conditional random variable $Y|X\sim U(0,1)$, how can I get this conditional ...
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1answer
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Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
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3answers
260 views

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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Maximum likelihood estimators of $\theta$ in $U(2\theta-1,2\theta+1)$ distribution

I understand why (D) is one of the answers but i dont know about the rest?
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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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1answer
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Joint cumulative distribution of independent random variables

X,Y,Z are non negative random variables which are independent and uniformly distributed in [0,1] and let $\alpha$ be a given number in [0.1]. Now how to compute $\text{Pr}(X+Y+Z>\alpha \;\;\; \&...