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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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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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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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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83 views

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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187 views

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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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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144 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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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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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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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 \;\;\; \&...
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The distribution of a posterior predictive p-value under certain assumptions

I am wondering if anyone can check my understanding of the following passage concerning posterior predictive p-values in the textbook "Bayesian Data Analysis 3rd Edition" on page 151: In the ...
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80 views

Standard Error of a function of ML estimators

The background of the problem is as follows: Suppose $X_1,...,X_n \sim U(a,b)$ independently where $a$ and $b$ are unknown parameters and $a < b$. Let $\hat\tau$ be the MLE of $\tau$, where $\tau =...
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Check to see if sample data could be uniformly distributed

I have data and I want to see if it is plausible that it comes from some uniform distribution. Is it uniformly distributed?
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Deriving a distribution whose pdf has the shape of a square + a triangle (a right trapezoid)

I want to the derive the PDF which looks like the sum of a triangular and uniform distribution which looks like this: To do this I have simply added the PDFs for the rectangular and triangular parts, ...
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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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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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Find $P(A^2 < B)$ where $A$ and $B$ are independent and uniformly distributed $\mathrm{Unif}(0,h)$, $h > 0$

I solved it two ways and in both the cases the answer is different and different from the actual answer. Approach 1: Since, $A$ and $B$ are independent, we can find the joint distribution of $AB$ ...
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Measure that takes samples that is minimized in expectation for a uniformly-distributed random variable?

I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i \in (0,1), i\in\{1,2,...I\}$, and provides a measure of ...
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The pdf of a standard uniform random variable divided by constant [closed]

For a random variable $\frac{U}{a}$ where $U$ is a standard uniform random variable, I'm trying to determine the pdf. I'm not so sure what I'm getting is correct as I'm getting some funny results ...
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Bayesian classification using uniform distribution

I would like to ask you if my thought first and my answer then to the following problem is right. Suppose that I have a 3-class 1-dim classification problem where the classes $\omega_1, \omega_2, \...
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1answer
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What a tests do I use to show that two parts of observations come from a two-dimensional uniform distribution?

I have a two-dimensional uniform data. I have splitted the data on two parts with a ...
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1answer
247 views

How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$

How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$. Is it a function of the order statistics?
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1answer
56 views

Transform X to get Y such that Y has a Uniform(0,1) distribution

A random variable $X$ has the PDF $f_X(x) = \frac{x - 1}{2}, \ 1 < x < 3$ Find a monotone function $u(x)$ such that the variable $Y = u(X)$ has the distribution $Uniform(0,1)$.
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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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49 views

probability that matrix $2\times2$ of Random variables is Invertible

Let $X_1, X_2, X_3, X_4$ to be 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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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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2answers
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How to interpret the results of a chi square and p-values of a distribution uniformity test

I am reading about a method called consistent hash designed to distribute load among servers. The best case scenario would be a discrete uniform distribution where each server would get the same ...
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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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Biasedness of Uniform Distribution MLE

How do I show that the maximum likelihood estimator for uniform distribution on $[0, \theta]$ for a random sample of size $n$ is biased? I've calculated the MLE as $\max_i\{X_i\}$. Intuitively, we ...
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1answer
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How to specify uniform distribution with same properties as normal distribution?

What I mean is, is it possible to specify a uniform random variable $U$ with random parameters $a,b$, where $a=-b$, and are generated from some other distribution, such that the marginal pdf of $U(a,b)...
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Conditioning to derive the distribution of function of uniform random variables

After seeing this question here, I was genuinely curious if there was a way to derive this distribution. I've attempted it below using the CDF for $Z$ and conditioning on the value of $Y$. It is ...
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How to find distribution function of sum of 2 random variables that are uniformly distributed? [duplicate]

I am stuck with this tutorial question in one of my stats module and I would greatly appreciate some help: Let $X1$ and $X2$ be independent random variables with $a = 0$ and $b = 1$ i.e. $X1$ and $X2$...
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Getting variance of function of two uniform RVs [duplicate]

Have two independent RV's $X$ and $Y$ sampled uniformly from $[0,1]$ and $C = (X-Y)^2$. Want $V(C$). Rewrote as $V((X-Y)^2) = V(X^2) - 4V(X)V(Y) + V(Y^2)$ but that's too messy. Is it correct to write ...
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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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What is the distribution of (1-CDF)? [duplicate]

We know that the cumulative distribution function (CDF) follows the $U[0,1]$ distribution. What is the distribution of (1-CDF)? Is it also follows the $U[0,1]$ ? (I believe it's true for the normal ...
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361 views

How to sample uniformly from the surface of a hyper-ellipsoid (constant Mahalanobis distance)?

In a real-valued multivariate case, is there a way to uniformly sample the points from the surface where the Mahalanobis distance from the mean of the is a constant? EDIT: This just boils down to ...
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104 views

Vector with elements from a uniform distribution, to be made unit

I have a two dimensional constant vector $\mathbf{A} = \left < 2,1 \right>$. Also, I have a vector $\mathbf{e} = \left < \epsilon_x, \epsilon_y\right >$. Both $\epsilon_x$ and $\epsilon_y$ ...
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69 views

Distance between angle distributions

I want to quantify the complexity of the street network of different cities. For each city I have the angle distribution of its streets. My hypothesis that the more complex the street network, the ...
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3answers
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An oddly skewed distribution of p-values

I stumbled upon an odd result which I have difficulties to explain. In the following code, $x_1$ and $x_2$ are very similar variables. Yet the distribution of p-values for the coefficient in $x_1$ is ...