Questions tagged [uniform-distribution]

The uniform distribution describes a random variable that is equally likely to take any value in its sample space.

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Properties of the diff of a sorted uniformly generated set [duplicate]

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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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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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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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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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
444 views

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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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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Relationship between two randomly-generated variables

Using stata, I generate two random variables and regress them with each other. ...
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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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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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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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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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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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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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Random variable is parameter for distribution of another random variable

What would you do to find probability when a uniform random variable is the parameter for the distribution of another uniform random variable. ie: $Z \sim Unif(0,1)$ $Y \sim Unif(0,Z)$ And we are ...
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Continuous random variables - probability of a kid arriving on time for school

A father leaves his home to take his son to school between 6:15 AM and 6:45 AM and it usually takes between 30 and 40 minutes to arrive there. Let $X$ and $Y$ be independent and random uniformly ...
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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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Inverse transform method on MCMC generated uniform draws

I understand that it sounds like why would anyone do this, but are there any references that use the inverse transform method to draw correlated samples from a distribution $F$ using MCMC samples from ...
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Theoretical justification behind assuming that the data is locally uniformly distributed, as seem to be used by manifold learning community

In at least three or more papers I've been studying that introduced novel algorithms for the estimation of intrinsic dimensionality (ID) based on nearest neighborhood (NN) techniques, I observed that ...
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Are arrivals uniformly distributed over a time interval when inter-arrival times are poisson distributed

Assume a fixed time horizon $T$ and suppose that inter-arrivals times at a queue (during the time horizon $T$) are poisson distributed with arrival rate $\lambda$. Will the arrivals be uniformly ...
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If $X=\sin\Theta$ and $Y=\cos\Theta$ with $\Theta$ uniformly distributed, how can I compute the joint pdf of $(X,Y)$?

I have a random variable $\Theta$ uniformly distributed between $[-\pi ,\pi]$, two functions $X=\sin\Theta$ and $Y=\cos\Theta$. I know that $X$ and $Y$ are uncorrelated but not independent. I want to ...
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Distribution function of $1/X$ when $X$ is uniform on $[-1,1]$

(from The Probability Tutoring Book, C. Ash, p. 157) Find the density of $Y$ if $Y = 1/X$ and $X$ is uniform on $[-1,1]$. The distribution function given in the answer key is $$ F(y) = \begin{...
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Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...
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Intuition behind generating discrete random variables from a uniform random number generator

The following is an exercise from Rice's Mathematical Statistics and Data Analysis: This problem shows one way to generate discrete random variables from a uniform random number generator. Suppose ...
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What does it mean for the uniform prior? [closed]

I wonder about the meaning of uniform prior of an unknown parameter. Any argumentation with detail explanation would be much appreciated.
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PDF of cosine of a uniform random variable with additional shift

I need to calculate the PDF of a random variable, which is quite similar to what was asked here. However, I have to deal with a shifted cosine function. Thus, my random variable is defined as $$Y:=cos(...
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1answer
391 views

Help with the posterior of a uniform distribution with a parameter that is uniformly distributed

Here is the question: My main issue is with the marginal distribution of θ, I know that the sampling distribution is 1/(θ^n), but what interval do we integrate on, it can't be [0, 1] because that ...
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What distribution does the mean of a random sample from a Uniform distribution follow?

For example, let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=1,\theta-1/2 < x < \theta +1/2$. Clearly, $X_i \sim U(\theta-1/2 , \theta +1/2)$. Some intuition would suggest that $\bar{X}...
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Sum of exponential of uniform random variables?

Let $F_{i}$ and $\phi_{i}$ are uniformly distributed independent random variables in the range $[-50,50]$ and $[-\pi/4,\pi/4]$, respectively. If $N = 10$ and $$Z = \sum_{i=0}^N e^{j(F_{i}+\phi_{i})}...
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How to estimate the PDF of the logarithm of a uniformly distributed random variable?

This is a question I have to solve and need help with. I know it's usual to give pointers and hints so the OP can follow from there. Thus, I'll appreciate all input that shows me the way to go. Let $...
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1answer
149 views

Computing $\mathbb{E}(S_n)$ and $\mathbb{V}(S_n)$ for Bernoulli data with a uniform probability parameter?

Take $U \sim \text{U}(0,1)$ as an underlying probability and generate $X_1,X_2,...,X_n \sim \text{Bern}(U)$ independent Bernoulli trials with this probability. The number of successes in the sample ...
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1answer
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Does runif (R) ever return 0/1 [closed]

The title says it all. Can it happen that runif (with bounds 0 and 1) returns 0 or 1 in R?
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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; \mathbf{y})...
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187 views

How to find confidence interval for Uniform([a,1])?

Let $ U_1, \dots, U_n $ be a random sample of uniform distribution over $ [a,1] $. Construct a confidence interval for $ a $ with $ 1-\alpha = 0.95 $. I managed to show that $ T = \min\{U_i\} $ is ...
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Is the t-test applicable when populations are sampled from uniform distributions?

I have two samples both taken from uniform distributions. The two samples differ in size. My first question is can I use the t-test to test the hypothesis that the two distributions are the same. Now ...
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probability distribution of a sum of random variables [closed]

Suppose we have a random variable $X$ $P[X=-1]=1/3$, $P[X=0]=1/3$ and $P[X=1]=1/3$ now let $Y=X^2$ we have $n$ independent realizations of $Y$ $(Y_1, Y_2,......, Y_n)$ what is the probability ...
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Using Hoeffding's inequality on sum of uniform variables

I have the following problem: $X_1,...,X_n$ are i.i.d. $\sim U(-3,5)$ continuous uniform variables in the support between -3 and 5. $S := X_1 + ... + X_n$. I need to use Hoeffding's inequality to ...
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What is the marginal probability distribution of the surface uniform sphere

I need to complete the following problem Let $~(X_1, X_2)~$ be a continuous random vector, with uniform density on the unit sphere $~\{(x_1,~ x_2,~ x_3) ~∈~ \mathbb R^3 ~:~ x_1^2 + x_2^2 + x_3^2 = 1\}...
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Approximate covariance of a uniform closure

I am interested to find a formula for approximate covariance of a uniform closure U(0,1)/summation{U(0,1)} for n independent U(0,1). I could derive an approximate expression for the variance = 1/(3n^2)...
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Can the difference of random variables be uniform distributed? [duplicate]

Given two random variables X and Y with some distribution D, is it possible to choose a D such that Z = X - Y is uniform? Is there a standard distribution D that would satisfy this?
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Uniform random variable is greater by a constant from another uniform random variable

I am trying to formulate the following question. X and Y are IID , uniform r.v. with ~U(0,1) What is the probability of P( X-Y-0.5 > 0) = ? 0.5 is a constant here and can be different. I do ...
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Is there a continuous function that accepts a single uniform random variable and returns two independent uniform random variables?

I can define a function $f(X) = (Y_1,Y_2)$ that accepts a random variable $X$ with a uniform distribution on $[0,1]$, and returns two independent uniform random variables $Y_1,Y_2$. This function ...
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Consecutive differences of a uniform law

Let $N>0$ be the number of considered samples. We draw $x_1, \ldots, x_n$ from a uniform distribution over $[0;1]$. We compute $y_1, \ldots, y_{n-1}$ the differences of the sorted $(x_i)_i$. I'd ...
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Expected frequency differences when sampling from a uniform?

Imagine I put people into different groups based on a uniformly distributed random variable $y = f(x)$ (e.g. microseconds of their arrival to a website). After a while, I observe how many people are ...
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1answer
39 views

Transformation of Uniform(0,1) random variable [closed]

Let $X \sim U[0,1]$. Find the pdf of $Y=4\sqrt{X}(1-\sqrt{X})$. I have been studying transformation of random variables and came across this exercise. Can anyone provide me a hint on how to ...
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1answer
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Sampling small set of indices from a very large set

I have a set containing 100 millions of indices. In each iteration, I choose $k$ sub-samples from this set, and once I select the $k$-samples, I increase or decrease the probability for the selected ...

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