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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For which value of $i$ is $p_i$ the largest? [closed]

Suppose $X$ is distributed uniformly on $(−1,1)$. For $i = 0, 1, 2, 3$, let $p_i = P\bigg(X^{2} \in \bigg(\frac{i}{4}, \frac{i+1}{4}\bigg)\bigg)$. For which value of $i$ is $p_i$ the largest? (A) 3 (B)...
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Analytical closed-form solutions for entropy [closed]

The differential entropy of random variable $X$ is $$H(X)=-\int_{-\infty} ^{\infty} f(x) \ln f(x) dx$$ I know of two cases where there exists a closed-form solution for $H$: $H(X) = \ln(\sigma \sqrt(...
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Generate Uniform Random Variates with Constant Norm [duplicate]

How can one generate $k$ uniform random variates centered at zero, $X_1, X_2, ..., X_k$, given a constant Euclidean norm, $c =\sqrt{X_1^2+X_2^2+...X_k^2}$?
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Expected number of uniform draws to exceed a first uniform draw

I came across the following problem (Problem number 27 from here): Aaron samples from the Uniform(0,1) distribution. Then Brooke repeatedly samples from the same distribution until she obtains a ...
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1answer
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Why does the distribution of the exponential random variable change to uniform distribution in this case?

I came across this very interesting question in a forum: If both X and Y are independent and exponentially distributed with parameter $\lambda$, find $E[X^2|X+Y]$ Someone gave the solution and ...
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Why do we need to triangulate a convex polygon in order to sample uniformly from it?

Suppose I want to uniformly sample points inside a convex polygon. One of the most common approaches described here and on the internet in general consists in triangulation of the polygon and generate ...
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1answer
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Maximum Likelihood Estimator of $\theta$ [closed]

Let $X_i$ be i.i.d $U(-\theta,2\theta)$ for i=1,2,...n. $f(x)=\frac{1}{3\theta}$ and $L(\theta)=(3\theta)^{-n}\mathbb{1}_{[-\theta<X_{(1)}<X_{(2)}<...<X_{(n)}<2\theta]}$. I don't know ...
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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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1answer
116 views

How do you uniformly sample spans from a bounded line?

Suppose you have a bounded and continuous line. For example, the line could include all real numbers between 0 and 3. How do you sample spans from the line such that... Any point on the line has an ...
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1answer
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Use t-distribution for sample mean of uniform RV's [duplicate]

Given there are 10 RV distributed by $U[0,\theta]$ ($\theta$ supposed to be uknown).I know sample mean ($\bar{X_{10}}$) of and sample variance ($\hat{s_{10}}$), can I found 95% CI for mean? My answer ...
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What's the expression for convolution of a uniform[a,b] density and a normal(0,d^2) density?

Suppose I have $X\sim Uniform[a,b]$ and $Y\sim normal(0,d^2)$, what's the expression for the density of $Z=X+Y$? Let $F_{Z}(z)$ be the cdf of $Z$ evaluated at $z$, and let $\Phi(\cdot)$ and $\phi$ be ...
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Bayes' Theorem Application

Ontario Public Health conducted a study on their test results for detecting SARS-CoV-2 (the virus that causes COVID-19) from Jan-April 20201. These are all patients who had symptoms and went to get ...
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1answer
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expected value of a fishing strategy

Suppose there is a pond with infinite number of fish. Weights of the fish are iid uniform $(0,1)$. We catch fish from this pond with the following rules: Each day we catch at most one fish from the ...
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2answers
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Is there a discrete distribution I can use for sampling in R?

Firstly, I don't have a stats background, so please accept my apologies for any errors or misunderstandings in the question below. I'm trying to use R to draw values from a discrete probability ...
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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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Does minimizing KL-divergence result in maximum entropy principle?

The Kullback-Leibler divergence (or relative entropy) is a measure of how a probability distribution differs from another reference probability distribution. I want to know what connection it has to ...
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3answers
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Generating random points uniformly on a disk [duplicate]

I have to randomly generate 1000 points over a unit disk such that are uniformly distributed on this disk. Now, for that, I select a radius $r$ and angular orientation $\alpha$ such that the radius $r$...
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Uniformly Distributed Residuals in Linear Regression

What can you say about your linear regression if the residuals are uniformly distibuted (and not normal)? I would like to consider the case I have a histogram showing residuals which are uniformly ...
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1answer
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Help me understand how to apply a beta-binomial model in order to estimate a parameter when there are several Bernoulli trials?

So, I have been presented with this question: A sample of 100 people were asked how many days they drove their car during the last week (inc. the weekend). The resulting frequency of response is shown ...
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Using a beta-binomial model to estimate the average for a uniform prior [duplicate]

Say we had a sample of 100 people who were asked how many days during the last week they drove their car. Let's say the resulting frequency table is as follows: Days, frequency 0, 1 1, 5 2, 3 3, 15 4, ...
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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 can I use transformation properties to obtain the distribution of $h(\mathbf{s})$?

Let that $\mathbf{s}=(s_1,s_2) \sim Unif(S)$, where $S$ is some spatial area. Suppose $y=h(\mathbf{s})=1-[exp(exp(\beta_0+\beta_1(\mathbf{s}-\mathbf{x})^T(\mathbf{s}-\mathbf{x})))]^{-1}$. We have that ...
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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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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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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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Collective risk model with parameter uncertainty

I was wondering if it was possible to get the EV and Variance of the sum of claims($S$) using a compounded distribution, given that: $$ N \sim Po(\theta) \\ \theta \sim U(0,0.1) \\ X \sim Exp(\...
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1answer
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How to show the rate of convergence of this maximum likelihood estimator is $n^{-1}$?

Suppose I have data $\{X_i\}_{i=1}^{n}\sim \text{Uniform}[0,\theta_0]$. We know that the maximum likelihood estimator of $\theta_0$ is $\widehat{\theta}=\max\{X_1,...,X_n\}$. How to show that $\...
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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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Sensitivity analysis with a uniform distribution

Is it possible to perform a sensitivity analysis on a uniform distribution? For more explanation: I can perform a sensitivity analysis on an exponential distribution by systematically varying the ...
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1answer
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Conditional Probability Uniform Bivariate Transformation Distribution

I'm reviewing probability theory from years ago and am a bit rusty. I'm not sure how to calculate the conditional probability for a uniform distribution after a bivariate transformation. Suppose X and ...
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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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Maximum likelihood estimator of $\theta$ for uniform distribution [closed]

I know that , For Uniformly Distributed random variables $X_1,X_2,\dots,X_n$ $\in \mathcal{R}$, the p.d.f is given by: $f(x_i) = 1/θ$ ; if $0≤x_i≤θ$ $f(x) = 0$ ; otherwise If the uniformly distributed ...
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1answer
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Find best unbiased estimator for $\theta$ when $X_i\sim U(-\theta,\theta)$

I am having an issue finding a best unbiased estimator for $\theta$. Any help is appreciated. Let $X_1, ..., X_n$ be a random sample from a population with pdf: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
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Expectation of differences between arcs on a circle

Consider a circle with a circumference of $n$. On this circle, I define two arcs of length $k<n$, $A_1$ and $A_2$. The centres of the two arcs are uniformly distributed on the circle. Let $\Omega_{...
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Question on a statistics symbol used in describing uniform distribution / likelihood function [duplicate]

When the function of a sample from a uniform distribution is described, there seems to be a symbol (either 1 or I) next to 1/theta. For example, the likelihood function of a sample from a uniform ...
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1answer
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finding PDF of Y, given Y|X [closed]

$$Y|X\sim Bin(X,n)$$ $$X\sim U([0,1])$$ How can I find the PDF of Y? I know that: $$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
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Confidence interval for simple sample, uniform distribution [duplicate]

Let $X_1, X_2$ be a simple sample from uniform distribution $U[0, \theta], \theta > 0$. Find constant $C$ such that an interval $(CX_{2:2}<\theta<\frac{1}{\sqrt\alpha}X_{2:2})$ is a ...
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If V=X+Y how to prove that there is no independent random variables X,Y make V a uniform distribution [duplicate]

If V=X+Y,how to prove that there is no independent random variables X,Y (with the same distribution function) make V a uniform distribution function on [0,1]
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1answer
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Density of square root of sum of squared independent uniform random variables [duplicate]

Let $X \sim U(-1, 1)$ and $X \sim U(-1,1)$. We want to find density function of $W = \sqrt{X^2 + Y^2}$. I got stuck and I have no idea, where I am making a mistake. This is my approach. Let $F$ be a ...
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Determine if clusters of combined values are coincidental

I have a set of measurement results like this (hex numbers): 30AF, 31AE, 30AF, 30AE, 31AE which vary only slightly, there are usually 3-4 distinct values per measurement set of 30 results(could ...
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Uniform distribution, estimates, MVUEs and Cramer Rao Lower Bound

As a revision exercise, I'm going through all of the distributions and deriving estimators. I've gotten to the $Uniform$. I've worked out the MLE and MOM estimators. The next step is to consider ...
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1answer
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Relationship between two randomly-generated variables

Using stata, I generate two random variables and regress them with each other. ...
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1answer
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how to write the joint density of two correlated uniform random variables?

Suppose the marginal distributions of $X$ and $Y$ are both $U[a,b]$, where $U[a,b]$ denotes uniform distribution on $[a,b]$. When $X$ and $Y$ are correlated, how to write their joint density for some ...
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1answer
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Why does Uniform distribution make sense?

This might be a dumb question, but I am suddenly confused on how to understand the PDF of a uniform distribution. For instance, the PDF of standard uniform is always equal to 1... How is that ...
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generating a uniform random variable from the first digit of an exponential random variable?

in "introduction to probability models", Ross talks about simulating with the rejection method, and he needs an exponential random variable, and a uniform random variable (used only for checking ...
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Independence for subsets of [0,1]

I'm struggling with this problem: Let $\mathbb{P}$ be the uniform measure on $[0,1].$ Define $A=(a,b)$ and $B=(c,d)$ with $a<c.$ State the necessary and sufficient conditions for $A$ and $B$ to be ...
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3answers
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Connection between uniform distribution on a set and uniform sampling from a set - intuitive pictures and necessary mathematical formulas

What's the connection between uniform distribution and uniform sampling from a set $S$? Are these two terms synonymous: "uniform sampling from a set" and "sampling from a set according to a uniform ...
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2answers
105 views

Ratio of Two Uniform Random Variables [duplicate]

If X1 X2 are independent Uniform variates on (0,1), Find the distribution of Z=X1/X2. I tried using the CDF method where P(X1<=zX2) is equal to z/2 when z is in(0,1). However, I am unable to find ...
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R: Problem with runif: generated number repeats (more often than expected) after less than 100 000 steps

After executing the code RNGkind(kind="Mersenne-Twister") # the default anyway set.seed(123) n = 10^5 x = runif(n) print(x[22662] == x[97974]) ...
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3answers
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Scale uniform interval estimator

I have a problem with the scale uniform interval estimator (Example 9.1.6, page 419, Casella-Berger). Let $X_1,\dots,X_n \sim \text{IID U}(0,\theta)$ be our observed data. We are interested in an ...

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