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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100 views

Uniform distribution with Gaussian Priors

Let's say i've got a uniform distribution defined as follows $$X \sim U[\min (\theta_1,\theta_2),\max (\theta_1,\theta_2)]$$ I've also got that $\theta_1,\theta_2$ are i.i.d zero-mean normal ...
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If A is distributed uniformly on [8,10] and B on [9,11], what is the probability that B<A?

I was asked this question in an interview, and did not initially answer correctly though I still think my interpretation may have been the correct one. The question was: There are two delivery ...
4
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1answer
399 views

Convolution for uniform distribution and standard normal distribution

Consider a random variable $U$ that has a uniform distribution on $(0,1)$ and a random variable $X$ that has a standard normal distribution. Assume that $U$ and $X$ are independent. Determine an ...
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Why is the sum of probabilities in a continuous uniform distribution not infinity?

The probability density function of a uniform distribution (continuous) is shown above. The area under the curve is 1 - which makes sense since the sum of all the probabilities in a probability ...
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109 views

$X_i \sim \text{Uniform}(0, \theta)$ iid; $Y = \max{(X_1,..,X_n)}$. Why is $\theta$ necessarily larger than $y$?

I'm going through Statistical Inference by Casella & Berger, and on page 419, in the intro section of interval estimation there is the following example (note: most of the text was left out as it'...
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188 views

General rule uniform distributed classes

Given a classifier working with double values e.g. between 0 and 1. There are two classes with different ranges. Their distributions are uniform, however, one class is more likely. Is picking always ...
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1answer
190 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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2answers
237 views

calculating correlation between binary vectors with generating with uniform distribution

I am working with some correlated binary files. I want to know, what is your opinion for calculating the correlation between binary vectors? for example, if I have two binary vectors X1 and X2 ...
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The probability of photon collision

I was reading a textbook and I couldn't figure out something that seemed really obvious: Assume that the space is uniformly and randomly filled with stars, and the mean radial distance between the ...
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2answers
350 views

What is the ratio of a N[0,1] and U[-1/2,1/2]?

I have come across a problem where I can reasonably assume that the numerator is a uniform distribution of the type U[-a,a], i.e., centered on zero, and the denominator is N[0,b]. This seems to be ...
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610 views

how to calculate the standard deviation of the sum of multiple uniform distribution

Given several independent uniform distributions, such as runif(0,5) and runif(0,50), how to calculate the sd of the sum of the two functions? And in what scenario would one needs to consider the sum ...
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How to determine if individual is related to a given population

I would appriciate your help in the following problem: I've been told that the bonobos height is uniformly distributed, with a,b equal to [70,76] respectivly. In this group, one individual (height=70....
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926 views

Sufficient statistics for Uniform $(-\theta,\theta)$

So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ? In other ...
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84 views

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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Expected distance of a stone thrown into a circular pond

A stone is thrown into a circular pond of radius 1 meter. Suppose the stone falls uniformly at random on the area of the pond. The expected distance of the stone from the center of the pond is: A) $1/...
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1answer
3k views

Confidence interval for mean of uniform distribution

I've been trying to compute a 95% confidance 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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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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145 views

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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102 views

Sufficiency and completeness of distribution

Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf $p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\ ...
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2answers
693 views

Confidence interval in U(0, $\theta$)

Let $X_n = X_1, X_2,..., X_n$ be a random sample of $X \sim U(0, \theta)$, where $\theta$ is an unknown parameter. Assuming confidence level $1 — \alpha$, find confidence interval for $\theta$ where: ...
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The relationship between UMVUE and complete sufficient statistic

Let $X_1,...X_n$ $U(-\theta , \theta)$ I want to find the UMVUE of $\theta$ if it is exists. My answer is , there is no UMVUE in this case. Because there is no complete sufficient statistic that ...
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76 views

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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PDF for the ith ORDERED uniformly random sample compared to an evenly spaced sample

Let $r_1 ≤ r_2 ≤ ... ≤ r_N$ denote an ORDERED set of N realizations of real numbers that are uniformly random on the number line from 0 to 1. Let $R_1 < R_2 < ... < R_N$ denote a set of ...
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Generate Beta distribution from Uniform random variables

I need to generate random numbers from Beta distribution using random variables from Uniform distribution. If I have two random variables $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$, and If $Y_1+Y_2&...
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1answer
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Derivation of a truncated posterior distribution

Let $T\sim~U[-t_0,t_0]$ and let $\epsilon\sim~U[0,1]$ and $T\perp \epsilon$. We define $R=T+\epsilon$. I was trying to compute $f(T=t|R=r,T\geq t_1)$, where $t_1\in[-t_0,t_0]$ is some pre-specified ...
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42 views

Construct joint distribution of $X,Y$ such that $E[X|Y=y,y\geq \bar{y}]$ is piecewise linear

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ (it would be great that $X$ also has uniform distribution) as long as it has ...
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257 views

Marginal distribution of spherical uniform

For a random vector $\mathbf{X} \in \mathbb{R}^n$ uniformly distributed on the surface of a sphere of radius $r$, the PDF is the inverse of the surface $$f_\mathbf{X}(\mathbf{x}) = 2\pi^{-n/2}\Gamma(n/...
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1answer
353 views

The mean of the max of two uniform distributions

What is the mean of max(U(0,1),U(0,1))? Judging by computer simulations, it must be at or around 2/3, but I have no idea how to compute the precise value.
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1answer
331 views

Finding complete sufficient statistic

Let $X_1 , ....,X_n$ be iid. $Uniform[-\theta,\theta]$. I need to find the complete sufficient statistic for $\theta$ or prove there does not exist such. I know that $T=(X_{(1)}, X_{(n)} )$ is a ...
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1answer
292 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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1answer
45 views

uniform pareto system error

Let $X\sim U(0,\theta)$. Given a sample of size n, the likeliohood function is $l(\theta \mid x)=\frac{1}{\theta^n}$ Consider a pareto prior distribution $\theta\sim pareto(k,a)$ with density $\frac{...
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1answer
248 views

Clarification: Bernoulli random variable with uniform distribution

Let $Z$ be a random variable which takes the value 1 when $U \le \frac 14$, $0$ otherwise, where $U$ ~ $\text{Uniform}(0,1)$. So $Z$ is a Bernoulli random variable with PMF $$p_Z(z) = \begin{cases} p,...
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40 views

Expected value $=\infty$? [duplicate]

If we let $U_1, U_2, U_3,..., U_n$ be uniform (0,1), find $$\mathbb E[\sum_{i=0}^n iU_i^{i-1}]$$which, using the linearity of expectation, gives $$\sum_{i=0}^n \mathbb E[i U_i^{i-1}]$$ Doing this ...
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1answer
480 views

Transformation of a uniform distribution

I'm sure this is a trivial problem, but I am having trouble committing to an answer. If $U$~$(0,1)$, then $Y=-ln U$. I know that $\mathbb P(Y \le y)$ = $\mathbb P(g(X) \le y)$ =$\mathbb P(-lnU \le ...
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Distribution of relative proportions of draws from two uniform distributions

Drawing samples from 2 uniform distributions and calculating pairwise proportions of drawn values gives rise to a distribution I cannot recognize. What distribution is this? ...
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2answers
301 views

Uniform Density Function

As we know the uniform probability density function is f(x)=1/(b-a) if i find the density function and area of this uniform distribution between (0, 1/2) then it would be f(x)=1/(1/2-0) f(x)=2 ...
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2answers
288 views

UMVUE estimates of uniform distribution mean and width

Given are the uniformly distributed samples $$x_n \overset{\text{iid}}{\sim} \mathcal{U}\left(\mu-\frac{w}{2}, \mu+\frac{w}{2}\right)$$ for $n = 1 \ldots N$.Then the UMVUE estimates of $\mu$ and $w$ ...
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2answers
135 views

Performing a uniform crossover of parents

I'm trying to do a crossover of two vectors in my program. There are two vectors, each a parent in a sense, and they're both a series of random binary digits (1's and 0's). My goal is to randomly ...
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1answer
124 views

Expected minimum distance from a point with varying density

I'm looking at how the expected minimum Euclidean distance between randomly uniform points and the origin changes as we increase the density of random points (points per unit square) around the origin....
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1k views

Distribution of “p-value-like” quantities under null hypothesis

It is well established that p-values are uniformly distributed when the null hypothesis is true. This follows from the definition of a p-value The probability of observing a value (or more extreme ...
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2answers
152 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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82 views

project hash into a fixed number of buckets using salt

We currently generate hashes for user-ids in our system (using md5) and bucket them into range buckets (range is 0 to 100, buckets can be 0-30, 31-70, 71-100). This process works this way - we ...
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1answer
101 views

Generate vector in $\mathbb{Z}^3$ with fixed sum and uniform distribution

I need to generate 3 discrete random variables whose sum is equal to a specified value (fixed) and is uniformly distributed, however each component of the sum has specified bounds. For example, $$ ...
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1answer
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PDF of the maximum likelihood estimator of a uniform distribution

Suppose $ \{X_1, \dots , X_n \}$ is a random sample from: $$ f_X(x; \theta) = \frac{1}{\theta} \text{, for } 0 \leq x \leq \theta $$ The Likelihood function is easy to calculate: $$ L_Y(\theta; y)...
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1answer
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guessing a 4-digit access code

The company I work for shares a building with another company, so when we need to access that building, we use a 4-digit access code (which never changed in several years) for the main door, and then ...
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325 views

Distribution of extreme values, case of uniform

Question: For $U_1 , \dots, U_n$ i.i.d. $U \sim \mathrm{unif}[0,1]$, we want to find the asymptotic distribution of $Z_n = n(1-U_{(n)})$ where $U_{(n)} = \max(U_1 , ... , U_n)$ I found this: ...
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2answers
733 views

Is there a continuous version of the Uniform distribution?

The Uniform distribution is not differentiable. Is there a differentiable distribution that approximates a Uniform?
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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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372 views

how do I draw samples from correlated uniform random variables given a correlation matrix

For a simulation study I am trying to generate samples from 4 correlated random variables following a multivariate uniform distribution, where all marginals are uniform variables and the population ...
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1answer
69 views

Measuring how well data is evenly dispersed

I am trying to measure how well scores from members of a group are dispersed over possible values. Think of it as a measure of diversity function. Thus, ...