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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10
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1answer
4k views

What is the benefit of using permutation tests?

When testing some null versus alternative hypotheses by a test statistic $U(X)$, where $X = \{ x_i, ..., x_n\}$, apply the permutation test with the set $G$ of permutations on $X$ and we have a new ...
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2answers
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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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4answers
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Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

For a uniformly distributed variable between 0 and 1 generated using rand(1,10000) this returns 10,000 random numbers between 0 and 1. If you take the mean, it ...
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1answer
62 views

Example: Writing the joint PDF $f(x, y)$ as the product of a marginal and a conditional probability function

I am presented with the following notes on Bivariate distribtions: If we can write the joint probability density function $f(x, y)$ of a pair of random variables $(X, Y)$ as the product of a marginal ...
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1answer
40 views

Maximum likelihood estimator in Uniform distribution [closed]

For Random sample with uniform distribution in Tetha< x< Tetha +1 What's the maximum likelihood function how can we maximize it?
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2answers
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Estimating parameters for the product of a lognormal random variable and a uniform r.v

Suppose I have a random variable which I suspect is the product of a lognormally distributed random variable $X$ and an independent uniformly distributed variable $U(0, 1)$. (The variables are the ...
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2answers
10k views

Log-uniform distributions

I am having some difficulty understanding what log uniform distributions are. Suppose that $\log X$ is uniformly distributed on the interval $[1,e]$. How do I describe $P(X=x)$? It seems like there ...
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1answer
348 views

Uniform distribution inside Log

What is the meaning of putting uniform distribution inside log? See page 5 of this paper (Corentlin et al.) To make it more clearer, within my knowledge, I think I should put a single value inside ...
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2answers
217 views

A data-independant transformation to discretize a range of values non-uniformly

I am sure this is trivial, but I am looking for a transformation that nonuniformly discretizes all values of a range into several bins. The bins should be variant and I'd like them to be smaller ...
2
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1answer
2k views

Variance of Estimator (uniform distribution)

In my script for statistical signals, I have some troubles to get the same result for the variance of an estimator $T$. Here is the example: Given the observations $X_1, \dots , X_N$ of a uniquely ...
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1answer
95 views

Pdf of log of ratio of uniform random variables

Suppose $U$ and $V$ are iid $U(0,1)$. I am trying to find the pdf of $W=log(U/(1-V))$. My approach is to suppose $Z = U$ and find the joint density of $(W,Z)$ and get the marginal of $W$. The pdf of $...
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1answer
40 views

What does add Gauss Random Noise mean?

I'm reading a article about LSSVM and in numerical part they created an artificial example and it says: The two features of samples are uniformly distrubuted in $[0,1)$ and $(1,2]$ with 50% of Gauss ...
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1answer
319 views

Finding maximum likelihood estimator, symmetric uniform distribution

Let $X_1, ...X_n$ be IID random variables with uniform$[ -\theta , \theta ]$ . I need to find the Maximum Likelihood estimator (MLE) of $\theta$. My work is as follows, The likelihood function is , ...
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1answer
66 views

What is the probability distribution for the squared distance between random points in an $n$-dimensional hypercube?

I choose random points $X,\,Y$ in $[0,\,1]^n$ (I assume all $2n$ Cartesian coordinates are $U(0,\,1)$ iids). What is the probability distribution of $\left\Vert X-Y\right\Vert _{2}^{2}$? Even the $n=1$...
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1answer
226 views

Uniform Prior on Normal Mean with Known Variance Implies Truncated Normal Posterior?

Let's say I have a uniform prior $\mu \sim \mathcal{U}(a,b)$, a normal likelihood $y|\mu \sim \mathcal{N}(\mu,\sigma^2)$ with known variance $\sigma^2$, and one observation $y$. Is then the posterior $...
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2answers
194 views

How to generate a uniform random variable from [1,7] if given a uniform[1,5] RV generator? [duplicate]

I saw this as a question on glassdoor and I've seen similar questions elsewhere. Can someone explain the intuition of how to solve a problem like this? There are two scenarios The uniform RVs in ...
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8answers
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Is there a plateau-shaped distribution?

I am looking for a distribution where the probability density decreases quickly after some point away from the mean, or in my own words a "plateau-shaped distribution". Something in between the ...
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2answers
190 views

Survival in two period game: mean of z|z<v with z=xy, x~U(a,b) and y~U(c,d)

I am looking for the functional form to describe the following: A random shock $x\sim Uniform(a,b)$ is multiplied with a second shock $y\sim Uniform(c,d)$. What is the mean value of all combined ...
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1answer
202 views

Spearman $\rho$ as a function of Pearson $r$

It is common to talk about the linear correlation, Pearson's $r$, between two random variables $\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$ as having two components: a) the copula and b) the marginal ...
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1answer
53 views

Reference books on uniform spherical distributions in multiple dimensions [duplicate]

QUESTION What is a citation of a book whose scope includes the uniform distribution [1] that is generalized to an $n$-ball [2]? Among other things, I'd like to read a book that include such ...
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1answer
636 views

Calculating the sum of dependent uniform random variables

My question derives from Problem calculating joint and marginal distribution of two uniform distributions. So, suppose we have random variables $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as ...
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1answer
34 views

What is the probabilty that X > 2 conditioning on Y = 2? (Homework)

another homework question here. Let 𝑌 be a binomial random variable with 10 number of trials and 0.2 probability of success. Let X be a uniformly distributed random variable over the interval [0, 3]. ...
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0answers
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How many classmates does a freshman have?

The freshmen at East China Normal University has just received their student ID. Let the last three digits of a student ID be ABC, then A is the class he is in, whereas BC is his number in the class. ...
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3answers
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How is $\theta$, the polar coordinate, distributed when $(x,y) \sim U(-1,1) \times U(-1,1)$ and when $(x,y) \sim N(0,1)\times N(0,1)$?

Let the Cartesian $x,y$ coordinates of a random point be selected s.t. $(x,y) \sim U(-10,10) \times U(-10,10)$. Thus, the radius, $\rho = \sqrt{x^2 + y^2}$, isn't uniformly distributed as implied by $...
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2answers
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What is cov(X,Y), where X=min(U,V) and Y=max(U,V) for independent uniform(0,1) variables U and V?

Let $X=\min(U,V)$ and $Y=\max(U,V)$ for independent uniform(0,1) variables $U$ and $V$. What's the covariance of $X$ and $Y$? Could you develop some calculations, especially regarding the computation ...
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1answer
40 views

What is the difference of $\Sigma$ esimation of Gaussian Copula based on known CDFs VS unknowns

Recently, I read this web page which explains the Copula package in R. A question occurred to me. Consider a data set $D_{n\times d}$ which $n$ is the number of samples and $d$ is the number of RVs. ...
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0answers
185 views

Are these equivalent (for p-values): super-uniform, stochastically larger than / dominating the uniform, conservative?

In the literature and online, I have found three different wordings that I think refer to the same concept: stochastically larger than uniform (which I take is ...
3
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1answer
64 views

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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1answer
117 views

Expected min distance between N Uniform RVs

Inspired by this recent question, what is the expected value of the minimum of the pairwise distances between $N$ uniform and independent RVs (uniform in $[0,1]$)? i.e. Let $X_1,...,X_N$ these RVs, ...
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2answers
421 views

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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1answer
139 views

Sufficient Statistic of Uniform $(-\theta,0)$

Let $X_1, ... , X_n$ be i.i.d random variables Uniform $(-\theta,0)$ , with $\theta > 0$ parameter \begin{align}f_{\theta}(x_1,x_2,\cdots,x_n)&=\prod_{i=1}^nf(x_i;\theta) \\&=\frac{1}{(\...
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0answers
43 views

Failed to get Uniform distribution from Gamma distribution

I read in Chapter 6 in this book that $p(K)\propto 1$ is equivalent to $e^{-K}\sim Gamma(0,0)I(0,1)$ where $K$>0 and is uniform distribution, e.g., $K \sim Uniform(0,100)$; $I(a,b)$ is the ...
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0answers
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The joint distribution of two different sums of the same independent uniform random variables

Given $k$ independent, uniform random variables $X_i \sim U(-a,a)$, $i=0,\dots k$, and two sets of coefficients $\{\alpha_i\}$ and $\{ \beta_i\}$, let $U = \sum_{i=1}^k \alpha_i X_i$ and $V = \sum_{i=...
4
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1answer
96 views

Transforming a uniform random variate to points on a circle

Sample $U \sim \text{Uniform}(0,\sqrt{2}-1)$. Accept $U$ with probability $1/(1+U^2)$ (else reject and sample again). Set $X = 2U/(1+U^2)$ and $Y = 1-UX = (1-U^2)/(1+U^2)$. With probability 1/2, ...
2
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1answer
93 views

How can I generate 2 sets of variables from different distributions with a correlation between them in r? [duplicate]

I am working in R and would like to generate 40 numbers from $\mathrm{N}(0,1)$ and another 40 from $\mathrm{Uniform}(0,2)$ with a negative correlation (for example: $r = -0.45$) between them. The ...
5
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1answer
389 views

Probability density of compound triangular distribution with uniformly distributed mode?

What are the probability density function and cumulative distribution function of a compound triangular distribution with uniformly distributed mode, both supported on $(-a, a)$? I.e., $$ m \sim \...
0
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1answer
55 views

Using an Autoencoder with uniformly distributed data

Setup: Dataset: 40k uniformly distributed 13-dim samples (floats between 0 and 1) AutoEncoder: (input: 13dim) - fc layer 13 dim, relu - latent layer - fc layer 13 dim, relu - (output: 13dim) Loss: ...
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2answers
579 views

Order statistics for non i.i.d. variables

$u$ is a random variable with uniform distribution in $[a,b]$ and is not observed (the distribution is known though). At every period $t \in \{2, ..., T\}$, the researcher observes $Y_{t+1} = \alpha ...
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1answer
88 views

Finding bias of $\hat\theta=\max\{x_1,\ldots,x_k\}$ where $x_i$'s are discrete uniform

I am working through some textbook problems and came across a problem I am having difficulty with. The problem asks to give the bias of a point estimate, namely for a given set of data $X = \{x_1, x_2,...
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0answers
50 views

Limiting Distribution of $n\left[Y_n\right]$ where $Y_n$ is the minimum of a sample of size n from Uniform$\left(0,\theta\right)$ distribution

Suppose $X_1,X_2,\dots,X_n$ is a random sample from Uniform$(0,\theta)$ for some unknown $\theta > 0$. Let $Y_n$ be the minimum of $X_1,X_2,\dots,X_n$. (a) Suppose $F_n$ is the CDF of $nY_n$. Show ...
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1answer
37 views

Hypothesis testing of sum of uniform distribution

I want to have some clarifications related to below question from Casella Berger Regarding $\phi_1(X_1),$ I understand that $\alpha$-value is 0.05 and hence we are looking for a value of $C$ for ...
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1answer
80 views

Prove that $X_{(n)} - X_{(1)}$ is an ancillary statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be an independent and equally distributed random sample whose distribution is uniform on the interval $(\theta,\theta+1)$, $-\infty<\theta<+\infty$. Then consider ...
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0answers
34 views

Estimating parameters of multivariate regression using the maximum likelihood method with a uniform distribution of residuals?

Representation of regression in a matrix form: $$Y = XA + E,$$ where: $X$ - independent variables, $Y$ - dependent variables (observations), $E$ - errors, which have a uniform distribution, $A$ - ...
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1answer
78 views

How to check if a distribution is different from uniform distribution using the values only?

I have a discrete empirical distribution, where the values D1 = {1,2,3,7,9,15,16,17,18,19,20} occurs with some different frequencies. The frequencies are not of interest to me. I am interested in how ...
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3answers
4k views

Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

This is an interview question for a quantitative analyst position, reported here. Suppose we are drawing from a uniform $[0,1]$ distribution and the draws are iid, what is the expected length of a ...
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3answers
933 views

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<\...
18
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4answers
411 views

Draw integers independently & uniformly at random from 1 to $N$ using fair d6?

I wish to draw integers from 1 to some specific $N$ by rolling some number of fair six-sided dice (d6). A good answer will explain why its method produces uniform and independent integers. As an ...
2
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1answer
85 views

The joint pdf of sample maximum and sample mean for uniform distribution?

Assume $$\{X_i\}\stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{Uniform}(0,1)$$ Find the joint p.d.f. of $$X_{(n)} \hat= \max \{X_1,X_2,\ldots,X_5\}\quad\text{ and }\quad \bar X\hat=\sum^n_{i=1}{X_i}$$ ...
1
vote
0answers
597 views

Histogram bin size to show deviation from uniform distribution [duplicate]

Simple question: Is there a rule of thumb for number of bins in a histogram with a uniform distribution? Details: I have a stochastic computer simulation that produces, as a test, $n$ values that ...
3
votes
1answer
238 views

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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