Questions tagged [uniform-distribution]

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

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Test for uniform distribution with multiple participants

I know how to use e.g. the Chi-Square test to check whether data are uniformly distributed. I was wondering whether the approach changes if the data comes from multiple participants, i.e., there is a ...
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UMVUE of $\theta$ for $\mathrm{Uniform}(0,\theta) $ where $\theta \in[1, \infty)=\Theta$

Let $X_1, \ldots, X_n$ be a random sample from $\mathrm{U}(0, \theta)$, where $\theta \in[1, \infty)=\Theta$, say. Here I tried to find complete-sufficient statistics for $\theta$ as my main target is ...
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Testing whether a set of points on the unit sphere is uniformly distributed

The canonical way to do the test is to perform the spherical harmonic transform of the empirical distribution and then check that the power spectrum decays, but this is presumably fairly expensive. Is ...
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Bayesian estimation of iid sample from Uniform$[0,\theta]$ and a Pareto$(\alpha,\beta)$ prior for $\theta$

I am working on Bayesian estimation: suppose that $X_1,\dots, X_n$ is an iid sample from Uniform$[0,\theta]$. Assume a Pareto prior for $\theta\sim Pareto(\alpha,\beta)$, i.e. $$ f(\theta)=\frac{\...
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Generating uniformly distributed random solutions of a linear equation

Given $n+1$ variables $p_0, p_1, \ldots, p_n$ defined over $\mathbb{R}^{+}$ so that $\sum_{i=0}^np_i=1$, and given a real number $1<x<n$, I want to generate random solutions of the equation so ...
2 votes
1 answer
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What metric should be minimized when searching from a subset of points that are as uniformly distributed across the space as possible?

Given a set of n points, I have to find a subset of given size m<n, so that the m points are as uniformly distributed as possible across the volume enclosed by the convex hull of set n. See example ...
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7 answers
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How to generate uniform distributed samples with given auto-correlation function

As I mentioned in the question title, I want to generate specific uniformly distributed samples. I need them to model a real world scenario. For my real data, I estimated a function, which ...
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1 vote
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Calculating the n-th moment of a RV, including negative fractional moments

I am stuck trying to solve the following exercise.. Let $X: \Omega\to [a,b] \subset \mathbb R$ be a uniformly distributed random variable. Compute the n-th moment of $X$, i.e. compute $\mathbb E[X^n]$ ...
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Transition probabilities in multi dimensional birth-death process

I've got an urn problem that I believe can be nicely modeled as a birth-death process (I'm very new to markov chains, so maybe this is simply the wrong approach). Suppose we have $k$ urns each of ...
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Is there something like a uniform distribution bounded on two intervals (or a uniform distribution with a gap in between)?

Let's say that I want to have a distribution from which there is an equal probability to draw a number between -0.9 and -0.3 and also a number between 0.2 and 0.8. Is there a distribution which allows ...
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Use chi-squared to check is numbers uniformly distributed

I want to check if the numbers produced by my random number generator are uniformly distributed. My code is below - is the statistical approach correct? Disclaimers - this isn't homework, it's ...
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Convergence in probability of $n \cdot \min _{1 \leq j \leq n} X_j$?

Suppose $X_1, X_2, \ldots \sim U(0,1)$ and $X \sim \operatorname{Exp}(1)$, and $X_1,X_2, \dots , X$ are independent. Does it follow that $n \cdot \min _{1 \leq j \leq n} X_j \stackrel{P}{\rightarrow} ...
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1 answer
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Non-IID Uniform Distribution

$A$ is uniform (0, 2) and $B$ is uniform(1, 3). Find the Cov$(W, Z)$, where $W=\min(A,B)$ and $Z=\max(A,B).$ Since $WX = AB,$ then by independence of $A$ and $B$, $E(WZ) = E(A)E(B),$ so that $$Cov(WZ)...
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In statistics how does one find the mean of a function w.r.t the uniform probability measure?

I am unfamiliar in statistics. My knowledge is in pure mathematics. Suppose $n\in\mathbb{N}$, where $X$ is in the $\sigma$-algebra of Caratheodory-measurable sets such that $X\subseteq\mathbb{R}^{n}$ ...
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Is it possible to conduct a Chi-Square Test to determine the distribution of a binary uniform distribution?

Suppose a data with a large sample size has a binary categoric variable (i.e. True/False, etc) and it is assumed that the null hypothesis of the sample follows a discrete uniform distribution. Is it ...
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Confidence interval for continuous uniform distribution [duplicate]

If X is a random variable from the uniform distribution on the interval [0,θ], how would one construct a 95% confidence interval for the method of moments estimator of θ? I have found that and also ...
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Approximaion with uniform mixture density

Assume that a RV is drawn from a distribution with PDF $f(x)$. I would like to approximate this distribution as a mixture of infinitely many uniform distributions. Without loss of generality, assume ...
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Entropy of a Continuous and Uniform Random Variable

The distribution of a uniform r.v. X is given as follows: The entropy is therefore: This means that as $∆$ approaches infinity, so does the entropy. This also means that as $∆$ approaches 0, the ...
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9 answers
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A three dice roll question

I got this question from an interview. A and B are playing a game of dice as follows. A throws two dice and B throws a single die. A wins if the maximum of the two numbers is greater than the throw of ...
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Why can Uniform distribution be deduced from this indicator function?

Given $f(X_1 \mid X_1 + X_2) \propto \mathbf{I}(X_1 \leq X_1 + X_2)$ where $\mathbf{I}()$ is the indicator function, and $X_1, X_2$ are independent random variables following $expo(\theta)$ with $\...
3 votes
1 answer
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Is there a way to measure uniformness of points in a 2D square?

Questions : Say I have 1000 points, that are distributed in a $[0,1]\times[0,1]$ square. It is not uniformly distributed. For example I might have "clusters" of points within the square, and ...
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Conditional expectation of Uniform given sum of Bernoulli trials

Given: [] Find the conditional probability distribution of theta given Sn and compute the conditional expectation. I believe the distribution of Sn will be a binomial with mean ntheta and variance (...
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Nonexistence of UMVUE for non-constant function?

I tried to prove the problem: Suppose X $\sim \ U(\theta-1,\theta+1)$, $\theta \in \mathbb{R}$. Then there is no UMVUE for $g(\theta)$ unless $g$ is a constant function. Here is my attempt: Suppose $...
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Confidence Interval for a MOM Estimator of a uniform distribution

If $X$ is a random variable from the uniform distribution on the interval $(\theta, 2\theta)$, how would one construct a 90% confidence interval for the method of moments estimator of $\theta$? The ...
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Unbiased estimators for uniform distribution

I have a given question that I just cannot figure out. From what I can understand, both estimators would be unbiased (since e_1 is the sample mean and e_2 is an unbiased estimator for uniform ...
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Question regarding splitting up CDF function for absolute value or square root, example using uniform distribution

Hi lets say your CDF is from $Unif(-1,1)$ so $F(x) = (x+1)/2$ Its easy to understand how $$P(X<x) = F(x) \implies xP(X>x) = 1 - F(x)$$. But how do I breakdown $P(|X| < x)$ or $P(X^2 < x)$? ...
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KL divergence between gaussian with uniform prior

I have 2 normal distributions $\mathcal{N}(\mu_1, \mathbb{I}_d)$ where $\mu_1$ is a fixed vector in $\mathbb{R}^d$ and $\mathcal{N}(\mu_2, \mathbb{I}_d)$, where $\mu_2$ is $\mu_1 + V$, where $V$ is ...
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prove that $4I[X^2+Y^2 \le 1]$ is a conditional expectation of $6I[X^2+Y^2+Z^2 \le 1]$

Currently, I'm stuck on a problem concerning indicator function, conditional expectation and estimating $\pi$. The main point of my problem is to show that $4I[X^2+Y^2 \le 1]$ can be expressed as $E[6*...
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What does it mean for two random variables to be independent?

Say I have a random variable X, X~Uni(0,1). And for the parameters 0 < a < b < 1: Z = {a < x < 1} , Y = {0 < x < b} What does it mean for those two variables to be independent? I ...
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Test for uniformity of a dataset with many ties?

My goal is to identify images (2D matrix) which are approximately uniform (all same brightness). Since I don't know of any "natural" tests for uniformity like Shapiro-Wilks for normality, I ...
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3 votes
1 answer
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Expected Number of Good Pairs

This is a question I had in my interview: we have $N$ i.i.d Uniform$(0, 1)$ random variables. Define a good neighbor for $x_i$ as the point that is closest to $x_i$ in absolute value. We call a pair $...
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Optimal estimator

Let $X, Z_1,$ and $Z_2$ be independent random variables taking values in the set ${0, 1}$. $X$ is uniformly distributed in ${0, 1},$ while the distributions of $Z_1$ and $Z_2$ are such that if we ...
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1 vote
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EM algorithm on a mixture of two uniforms?

I am having serious issues understanding the EM algorithm, both the E and the M steps when it comes to a mixture of two uniform distributions. I am given the pdf of the mixture which is: $f(x)= \frac{...
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2 answers
498 views

Does a uniform random distribution become a normal distribution?

I have $N$ numbers from uniform random distribution. Now I have done a transformation on these $N$ numbers as: each $x$ is converted to $\frac{(x-\mu_N)}{\sigma_N}$. Where $\mu_N$ is the mean of $N$ ...
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2 votes
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KL-Divergence on Uniform Distribution, is this correct?

$P_1$ is uniform on $[0, 0.5], ~P_2$ is uniform on $ [0, 1] $ What is the KL-divergence $(P_1 \Vert P_2)?$ Attempt: $$ D(p\Vert q) = \sum_{x\in X} p(x) \log \frac{p(x) }{q(x) }.$$ KL$(P_1\Vert P_2) =$...
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Are there better measures of entropy

Related question here I am trying to measure the uniformity of multimodal distributions and am looking into using entropy. I would like a measure of entropy that is higher for the first distribution ...
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Why does somebody argue that the number of bootstrap replications should not be a multiple of 10?

At a recent conference somebody claimed that the size of the bootstrap replications should always be 999 rather than 1000. Which argument supports this claim?
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How to Justify this Two-Sided Test is UMP with NP Lemma?

UMP tests generally do not exist for two sided tests, ie $H_0: \theta = \theta_0$ vs $H_a: \theta \neq \theta_0$. However, if we observe $n$ iid observations of $X\sim Unif(0,\theta)$, we can ...
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2 votes
1 answer
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Joint Uniform Distribution Probability Problem

Let $X \sim U(0,1) $ and $Y \sim U(0,x) $. Calculate $$ \Pr(X >0.5 | Y= 0.25)$$ Is this a trick question ? Since $\Pr(Y = 0.25) = 0$, right ?
7 votes
1 answer
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General form of the distribution of distances to a fixed point in rectangle

The answer given to the question Probability distribution of the distance of a point in a square to a fixed point solves for the "distribution of the distance between the origin (0,0) and a ...
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Convergence of a function having a big summation at each sample

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ Here, $U_m$ samples are random numbers coming from a Gaussian distribution $$U_m \sim \mathcal{N}(\mu_u, \sigma_u)$$ and ...
1 vote
2 answers
67 views

Probability random variable is less or equal to k-th out of two samples when ordered

Given the random variable $X$, $\{X_{i}\}_{i=2}^{n}$, $\{Y_{i}\}_{i=2}^{n}$ all iid and lets denote $X_{(k)}$ as the k-th statistic of $\{X\} \cup \{X_{i}\}_{i=2}^{n}$ and $Y_{(k)}$ for $\{X\} \cup \{...
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1 answer
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QR interview problem Guessing order of draws from iid U(0,1)

This is for QR at two well know trading firms (think jane street, HRT, Citadel, Jump ...)(not BB bank). Question prompt: Given n iid Uniform distributed r.v.s. $x_i$ ~ U(0,1). $x_1$ is drawn first, ...
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2 votes
1 answer
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Extracting statistical parameters from a mixture of two distributions of different kind

I have a dataset b (as a list in Python) of length 100 I know that is amounts to the mixture of two distributions: A normal distribution A uniform distribution ...
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Does box muller algorithm produce independent and identically distributed samples? [duplicate]

Since, box muller generates two samples of standard normal distribution can we say that it produces IID samples
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Finding the MVUE of the center of a circle of unknown location

Is there a known analytic solution for finding the minimum variance unbiased estimator of a disk of an unknown location given that a sample of $n$ points was drawn uniformly and randomly from the disk ...
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3 votes
1 answer
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Conditional distances in order statistics

Assume I have $n$ points sampled independently from the uniform distribution on the unit interval. After ordering the sample I get the points $X_1, X_2, \dots X_n$ such that $X_1 \leq X_2 \leq \dots \...
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2 votes
2 answers
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probability that the players will exchange their initially drawn number

Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own ...
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1 answer
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Find the MLE density function of uniform [-\theta,\theta] [duplicate]

For $X_1,\dots,X_n$, i.i.d $X_n \sim \mathrm{unif}[-\theta,\theta]$, the ML: $\hat\theta_{MLE}=\mathrm{max}\{-X_{(1)},X_{(n)}\}$. Find the density function. Hint: For $x_1,\dots,x_n$ : $\textrm{max}\{-...
1 vote
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MLE of the Uniform Distribution

In a uniform distribution where $0\leq X \leq \theta$, the pdf is represented as $f(X|\theta) = \frac{1}{\theta}I(0\leq X \leq \theta)$, and the likelihood is $L(\theta) = \prod\frac{1}{\theta}I(0\leq ...
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