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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Circular Uniform Distribution Function Question [closed]

I have a question related to the cdf portion of the problem below: Problem. Suppose $W$ is uniformly distributed on $[0, 2\pi)$, and let $Z = (X, Y) = (\cos(W), \sin(W))$. What are the marginal ...
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test if one event is more frequent than another

I am observing 2 types of events with counts (A and B) and I want to know if one is more frequent than the other for the time that I am observing. I know the chance of an appearance is uniform over ...
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Continuous distributions function has uniform distribution [closed]

Let X be a random variable with continuous distribution function F (x) than it has uniform distribution ,i.e.,U (0,1). Please prove this theorem.
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Pdf of the sum of two independent Uniform R.V., but not identical

Question. Suppose $X \sim U([1,3])$ and $Y \sim U([1,2] \cup [4,5])$ are two independent random variables (but obviously not identically distributed). Find the pdf of $X + Y$. So far. I'm familiar ...
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Sobol variance based decomposition

I have 6 input variables, each of which is normally distributed. Can I use Sobol variance-based sensitivity analysis? I have read some articles where they said that input variables must have uniform ...
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Creating our own normal distribution function in R [closed]

I have a homework question which requires me to create my own standard normal distribution function which I had derived. Nonetheless, I have problem doing so. What I did was to have two independent ...
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1answer
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Normalize the posterior density for a Cauchy Distribution C ($\theta$,1) and a Uniform [0,100] prior [closed]

Using a Bayesian approach we have $$P(\theta|\text{data})= P(\text{data}|\theta) \frac{P(\theta)}{P(\text{data})}$$ Therefore, the posterior distribution will be proportional to $$\frac{1}{N} (1+(y+\...
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Domain problem when calculating marginal density

I have the following homework assignment: the life expectancy $X$ of a lamp has exponential distribution with rate $\lambda$. The rate depends on the production proccess, such that its population can ...
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1answer
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Irwin-Hall distribution scaling

From https://en.wikipedia.org/wiki/Irwin–Hall_distribution: The generation of pseudo-random numbers having an approximately normal distribution is sometimes accomplished by computing the sum of a ...
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1answer
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Showing convergence to a uniform distribution

I've been told that X is a random variable with a discrete uniform distribution on the set (1/n,2/n,...,1) and have been asked to show that it converges in probability to U(0,1) which is a uniform ...
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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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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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1answer
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Probability the next draw from a distribution is greater than some number given a previous draw

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. I am looking to solve for two different probability functions, though I think ...
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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
179 views

Sufficient statistics in the uniform distribution case

I am currently studying sufficiency statistics. My notes say the following: A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$, $$L(\theta; \...
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Algorithm for uniform sampling with bounded replacement

Is there a simple algorithm to sample from the uniform distribution on sequences of $n$ numbers, each taking one of $m$ integer values from $0$ to $m-1$, where each value can be repeated at most $r$ ...
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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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Is there an explanation for why there are so many natural phenomena that follow normal distribution?

I think this is a fascinating topic and I do not fully understand it. What law of physics makes so that so many natural phenomena have normal distribution? It would seem more intuitive that they would ...
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A question about a sum of squares of uniform random variables

For independent and identical $V_1,V_2\in U(-1,1)$, what is the probability that $V_1^2+V_2^2<1$? I tried but can't get an answer, the answer is $\frac{\pi}{4}$
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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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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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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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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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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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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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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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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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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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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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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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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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Why are p-values uniformly distributed under the null hypothesis?

Recently, I have found in a paper by Klammer, et al. a statement that p-values should be uniformly distributed. I believe the authors, but cannot understand why it is so. Klammer, A. A., Park, C. Y.,...
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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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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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1answer
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Why small values produce undulating densities when ploting logarithm of a loguniform prior (in R)?

I am using a program that draws random values in a log-uniform distribution let say between 1 and 100. When I plot the density of the produced values with R it looks like a log-uniform distribution ...
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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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Estimation derived from ignorance

Is something wrong with the following reasoning? Mostly I wonder how could one derive uniformly random arrival from ignorance. But even if that derivation is invalid generally, it seems reasonable ...
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1answer
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Simplest possible (uniform) sampling of the interval [0,1] with n points

The simplest possible sampling of a function in the region [0,1] for the purpose of finding the area under the curve, for instance, could be just take the left offsets of the bars. That is, you have ...
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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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When should I use the Normal distribution or the Uniform distribution when using Xavier initialization?

Xavier initialization seems to be used quite widely now to initialize connection weights in neural networks, especially deep ones (see What are good initial weights in a neural network?). The ...
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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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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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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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Test for uniformity in R

i am searching for a test for uniformity in R. ks.test(x,'punif') looks quite good, but my data has only 6 different values (results of rolling a die) which leads ...

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