# 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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### Show that $\min(U,1-U)$ and that $\max(U,1-U)$ are uniform

Let $U$ be uniform on $(0,\ 1)$. Show that $\min(U,\ 1-U)$ is uniform on $(0,\ 1/2)$ and that $\max(U,\ 1-U)$ is uniform on $(1/2,\ 1)$. I'm not sure how to approach... the only hint i have is that a ...
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### Convergence of $X_{{\lfloor n/3 \rfloor}}^ \space\small{(n)}$ if $X_1, \dotsc , X_n \sim U(0,1)$

$X_1,X_2,\dotsc ,X_n$ are independent, uniformly distributed random variables on the interval $[0,1]$ The question is the convergence of the sequence: $X_{{\lfloor n/3 \rfloor}}^ \space\small{(n)}$. ...
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### Measure for the uniformity of a distribution

I can't seem to find a well established and simple statistical measure of uniformity in occurrence datasets in the presence of zero-valued categories. I've looked at Shannon's entropy which seems to ...
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### Claims and questions regarding $n$-ball distribution?

CONTEXT In my research, I am utilizing an $n$-ball distributions along with two related distributions. I'd like to make certain I have a firm handle on the way to describe my three distributions. I ...
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### The Distribution of max(X,1/X)

If $X \sim \text{U}(0,1)$, what is the distribution of $Y = \max(X,1/X)$? I know for this particular problem, $Y = \max(X,1/X) = 1/X$, whose distribution can be easily attained directly. However, I've ...
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### CDF of Z=XY with X~Uniform(0.5,1.5) and Y~Uniform(0.8,1.5)

I am looking for the CDF of the product of two independent random variables (X and Y) with uniform distributions. Both random variables uniform distributions have interval boundaries (upper and lower ...
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### How to find the CDF of a random variable uniformly distributed around another random variable?

I'm working on some game theory models of incomplete information (which I've posted about a few times here). I think this question is pretty straightforward though, so the actual context is ...
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### What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables

$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs. ...
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### Random number generation distributed like a translated weibull from uniform random generator

If $X$ is uniformly distributed on $(0,1)$, then the random variable $\lambda(-\ln(1-X))^{1/k}\$, is Weibull distributed with parameters $k$ and $\lambda$. With this, I can get random numbers ...
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### Uniform Random on $(-\infty,\infty)$

Imagine picking a 1 when any real number is equally likely. What is the pdf? Does this idea have a known use? What is its name? There could be a use for a uniform random real number. It could end ...
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### Marginal of a uniform distribution

Given $f\left(x|\theta\right)=1/\theta, 0\leq x\leq \theta,L\left(\theta, a\right)=\left(a-\theta\right)^2,$ and $\pi\left(\theta\right)=\theta e^{-\theta},\theta\gt 0$ I've seen Problem calculating ...
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### Problems with extremum of two uniform random variables

Here is the problem from the book: Let $X = \min(U,V)$ and $Y = \max(U,V)$ for independent $\text{uniform}(0,1)$ variables $U$ and $V$. Find the distributions of a) $X$; b) $1-Y$; c) $Y-X$. I ...
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### Uniform Distribution Test

I've got a data-set which I assume is uniformly distributed. Say I've got N=20000 samples and a suspected p=0.25. This means ...
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### Uniform distribution presentations seem incomplete

I'm studying the book "A practical guide to quantitative finance interviews." This is how it presents the discrete uniform distribution: I've seen several other sources present it the same way (http:/...
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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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### Monte Carlo simulation of $\pi$

I am trying to find the value of $\pi$ using Monte Carlo simulation. However, I don't want to generate two random numbers as coordinates. Instead, I want to select a point on the edge of the square ...
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### Discrete uniform distribution vs. binomial distribution; what's the difference?

I read online that a uniform distribution gives to all its values the same probability to occur. In the discrete case, an example of this would be a coin flip. (as they have the same probability to ...
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### Geometric mean of uniform variables

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to chi-...
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### Covariance of a set of uniformly distributed unit vectors?

I have a set of uniformly distributed unit vectors within a "cone" (essentially a subset of a uniform distribution on the unit sphere, as described here). I've found how to get the covariance matrix ...
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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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### 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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### Convergence of sum of exponentially weighted random variables

I don't know if the title is accurate, but I have this problem: I have iid RVs $Y_k$ that has a value from {0,1,...,9} with equal probability. I need to show that $$X_n = \sum_{k=1}^{n}Y_k10^{-k}$$ ...
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### What proportion of the space is taken up by independent discrete uniform variables

If you take $N$ independent uniform random selections from a discrete space with $M$ possibilities (with replacements), then what proportion of the possibilities will have been selected? Formally, ...
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### Showing Independence for X and frac(X + Y)

Suppose that we have independent samples $X_1, X_2 \sim \text{unif}[0, 1)^d$. I'm asked to show that $Y_1 = X_1$ and $Y_2 = X_1 + X_2 - \lfloor X_1 + X_2 \rfloor$ are also independent uniform samples ...
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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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### MLE discrete uniform distribution

Let $X_1, X_2, \ldots, X_n$ be a random sample of discrete random variable with Uniform distribution on set of integers $\{-\theta, -\theta+1, ... ..- 1, 0, 1, \theta-1, \theta\}$ where $\theta$ is ...
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### What is a good example of a non-informative prior for the uniform distribution?

I recently noticed that for non-informative priors, people usually use something like a uniform prior, which works for many different distributions. However, assuming that your likelihood is nothing ...
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### Stats test which test of the NULL that a distribution is uniformally distributed

I'm looking for a statistical test which tells the probability that a given sample comes from a uniform distribution. Shapiro test wether a sample comes from a normal distribution. I'm looking a ...
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### Probability of a random variable to be the largest among others

Let us have $N$ random variables generated by uniform distribution. That is, $$u_i \sim \mathcal{U}(0,1),\quad i=1,\ldots,N$$. What is the probability of $u_N$ being the largest? I.e., how can I ...
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### Uniform distribution & generation of extreme values in R

I'd like to generate a new point which should be uniformly distributed on the interval [a, b) (i.e. including the left extreme value - a and exluding the right extreme value - b). The ...
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### Testing implementation of Anderson-Darling test for uniform RV

I am trying to write unit tests for a whole mess of statistics code. Some of the unit tests take the form: generate a sample following a null hypothesis, use code to get a p-value under that null, ...
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### minimal sufficient statistic for $U(\theta, \theta+c)$. $(\theta,c)$ unknown

Suppose $X_1,\cdots,X_n$ are $i.i.d$ from a distribution with p.d.f $$\delta_{(\theta,c)}(x)=\frac{1}{c}\mathbb{1}_{(x\in[\theta,\theta+c])},$$ where $\theta\in\mathbb{R}$ and $c\in\mathbb{R}^+$ ...
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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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### 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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### 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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### Individual Bernoulli distributions from Discrete Uniform

Perhaps a silly question, but: Let's say I want to randomly pick a number between 1-5, so that each number is equally likely to be picked. In other words, the number I pick is discrete Unif{1,5}. ...
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### Detecting relationships between two sets of circular data

I have a set of points $(x_i,y_i)$ where each x & y value is circular & can take on a value from -pi to pi. (The topology of the data is a torus, but I am not sure how relevant that is to the ...
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### Any practical uses of inverse uniform distribution?

To motivate a paper in game-theory I need examples of real-life uses of the inverse uniform distribution (http://en.wikipedia.org/wiki/Inverse_distribution#Inverse_uniform_distribution). Which type of ...
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### Estimating upper bound of uniform distribution from max of sample

This is actually part of a problem from All of Statistics: $X_1, X_2, \ldots, X_n \sim \text{Uniform}(0, \Theta)$. And $Y = \text{Max}\{X_1,\ldots, X_n\}$. If you're given that $Y > c$, can you ...
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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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### Does $\cos(U)$ have the same distribution as $\sin(U)$, when $U \in (0, 2\pi)$?

Consider an uniformly distributed variable $U$ in $(0,2\pi)$. My impression is that $\cos(U)$ have the same distribution as $\sin(U)$. Is my assumption correct?
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### Expectation of roots of a quadratic equation

The quadratic equation $x^2 -ax+ b = 0$ is known to have two real roots, $X_1$ and $X_2$ $(X_1 > X_2)$ but the coefficient $b$ is a positive unknown and can be assumed to have a uniform ...
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### Distribution of X-U(0,1) conditioned on sigma algebra of Y/X, where is Y is U(0,1)?

The question I have is: Define X,Y to be two independent uniform(0,1) random variables and $Z:=\frac{Y}{X}$ Compute $P(X<x|\sigma(Z))$ The answer given apparently by "straightforward elementary ...
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### Check that a statistic is complete

I have a question regarding completeness of a statistic. So the problem is: $n$ numbers are chosen randomly and independently between $a$ and $b$ ($0 < a < b$) but the information about $a$ and ...
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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 ...
I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by  \pi_G(\xi|\mu,\sigma) = {1\...
### Where is the uniform distribution with one parameter ($U(\theta, k \theta)$) useful for modelling?
I recently came across the distribution $U(\theta, k \theta)$ (where k is known) in the context of statistical theory (as a nice toy example for finding MLE and the likes). However, I was wondering ...