Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [uniform-distribution]

For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

55
votes
1answer
1k views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{...
28
votes
2answers
29k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniform variables with a difference support, how should one proceed ?...
22
votes
2answers
469 views

Answered: With what probability do $4$ points placed uniformly randomly in the unit square of $\mathbb{R}^2$ form a convex/concave quadrilateral?

I have this problem that I've struggled with for a while. If you place $4$ points randomly into a unit square (uniform distribution in both $x$ and $y$), with what probability will this shape be ...
15
votes
0answers
239 views

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms. It suffices to show that the terms of the sequence $$\,b_n=\mathrm{e}^...
12
votes
0answers
327 views

Picking points on a sphere at random

Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that ...
11
votes
2answers
16k views

Probability the three points on a circle will be on the same semi-circle

Three points are chosen at random on a circle. What is the probability that they are on the same semi circle? If I have two portions $x$ and $y$, then $x+y= \pi r$...if the projected angles are $c_1$ ...
10
votes
2answers
6k views

joint distribution, discrete and continuous random variables

This may be trivial, but if X is a random variable uniformly distributed over $[0,1]$ and Y is a discrete random variable such that $\mathbb{P} (Y=y_1) = \lambda \in (0,1]$ and $\mathbb{P} (Y=y_2) = 1 ...
9
votes
2answers
10k views

Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
9
votes
1answer
8k views

Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
9
votes
1answer
329 views

Can the sum of two independent r.v.'s with convex support be uniformly distributed?

Is it possible to prove that the sum of two independent r.v.'s $X$ and $Y$ with convex support cannot be uniformly distributed on an interval $[a,b]$, with $a < b$? (Let us rule out the trivial ...
9
votes
0answers
505 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
8
votes
3answers
310 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
8
votes
1answer
6k views

Complete Statistic: Uniform distribution

Take a random sample $X_1, X_2,\ldots X_n$ from the distribution $f(x;\theta)=1/\theta$ for $0\le x\le \theta$. I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete. Now, I know I should ...
8
votes
2answers
3k views

Draws from the uniform distribution are taken until the sum exceeds 1. What is the expected value of the final draw?

I was thinking about this question after a related problem: what's the expected number of draws for the sum to exceed 1? For that problem, the answer is known and is a surprising result http://...
8
votes
1answer
7k views

Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
8
votes
2answers
130 views

Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$

Let $X_{1},\ldots,X_{n}$ be i.i.d. $U[\alpha,\beta]$ r.v.s., and let $X_{(1)}$ denote the $\min$, and $X_{(n)}$ the $\max$. Show that $$ \mathbb{E}\left(\overline{X}_{n}\mid X_{(1)},X_{(n)}\right) = ...
7
votes
2answers
6k views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
7
votes
2answers
4k views

Density of sum of two uniform random variables

I have two uniform random varibles. $X$ is uniform over $[\frac{1}{2},1]$ and $Y$ is uniform over $[0,1]$. I want to find the density funciton for $Z=X+Y$. There are many solutions to this on this ...
7
votes
0answers
538 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
6
votes
2answers
4k views

Probability of difference of random variables

How can I compute this probability? I do not know what to do since it involves two random variables. Let $X$ and $Y$ be uniform random variables on $(0,1)$. How can I compute this? $$ P(|X-Y| < ...
6
votes
3answers
7k views

Binomial distribution with random parameter uniformly distributed

I have a problem with the following exercise from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001 (page 155, ex. 6): Let $X$ have the ...
6
votes
2answers
108 views

Is there a fundamental reason to expect $e$ to appear in this probability question?

I came across the following question, whose answer is $e$. I was sort of amazed, since I didn't see a reason why $e$ should be making an appearance. So, phrasing the main question in the title another ...
6
votes
1answer
664 views

Order statistics for discrete uniform random variables

Let $X_i, i=1,\cdots,N$ be i.i.d. discrete uniform random variables, taking values in the range $\{0,1,...,M-1\}$. Let $X_{(i)}$ denote the $i$-th order statistic. What are the values of $\...
6
votes
1answer
82 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively need ...
6
votes
1answer
401 views

Average sine of an angle between two rays in a cone

I'm looking for an average value of sine of an angle between two rays, lying within a cone with a certain angle. Given a cone with an aperture of ${2\chi}$ and two rays lying within the cone. The ...
6
votes
1answer
126 views

Distribution of minimum of difference between uniforms $(0,1)$

Let $U_1,...,U_N$ be a sequence of independent uniformly $(0,1)$ random variables. Define $Y_k := U_k - U_{k+1}$. Find the distribution of $\min\left\{Y_k\right\}_{k=1}^{N-1}$. I've get the ...
6
votes
1answer
308 views

Improper Uniform Distribution on $\mathbb{R}$

It is not possible to write a function representing the uniform distribution, say $\mathcal{U}$, on the real line, $\mathbb{R}$ (in a Bayesian context, this is an improper prior). My question is: is ...
6
votes
0answers
190 views

Is this fraction undefined? Infinite Probability Question.

Where $\frac{1}{\infty}$ and $\frac{\infty}{\infty}$ are both undefined terms that generally lead to nonsense, I'm wondering if we can assert that...: $$\frac{1+1+1+\cdots}{1+1+1+\cdots} = 1$$ ...or ...
6
votes
0answers
86 views

$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...
5
votes
2answers
7k views

Distribution of $\max(X_i)\mid\min(X_i)$ when $X_i$ are i.i.d uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) \mid \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$ I agree that the probability ...
5
votes
2answers
5k views

Prove there exists no uniform distribution on a countable and infinite set.

Can anyone help me with this problem, I can't figure out how to solve it... Let $X$ be a random variable which can take an infinite and countable set of values. Prove that $X$ cannot be uniformly ...
5
votes
1answer
2k views

Proof that normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = \left(\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \...
5
votes
2answers
843 views

Meeting probability of two bankers: uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
5
votes
2answers
133 views

For i.i.d. $U(0,1)$ random variables $(X_i)$, $\max\limits_{1\le i \le n/2}\{(1-2i/n)X_i\}\to1$ in probability

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
5
votes
1answer
338 views

Expected determinant of a random symmetric matrix

The three distinct entries of a $2 \times 2$ symmetric matrix are drawn from the uniform distribution over $[-60, 60]$. What is the expected determinant of the matrix? I assume it is $0$ but I am ...
5
votes
4answers
506 views

The One-way Highway

This is supposedly a thought-provoking interview question asked, and I though I have an idea of a possible solution, I can't prove it. The question is the following: You have $n$ cars that are all ...
5
votes
2answers
2k views

Sum of discrete and continuous random variables with uniform distribution

Could you tell me how to find the distribution of $Z = X+Y$ if $X$ is a random variable with uniform distribution on $[0,1]$ and $Y$ has uniform distribution on $\{-1,0,1\}$? $X$ and $Y$ are ...
5
votes
3answers
17k views

Joint density problem. Two uniform distributions

This is the problem: An insurer estimates that Smith's time until death is uniformly distributed on the interval [0,5], and Jone's time until death also uniformly distributed on the interval [0,10]. ...
5
votes
1answer
603 views

Probability that, given a set of uniform random variables, the difference between the two smallest values is greater than a certain value

Let $\{X_i\}$ be $n$ iid uniform(0, 1) random variables. How do I compute the probability that the difference between the second smallest value and the smallest value is at least $c$? I've messed ...
5
votes
1answer
949 views

What's the distribution of the exponential of uniformly distributed variable?

I want to know the distribution of $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}[-\pi;+\pi]$. From the book "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai I ...
5
votes
2answers
78 views

Variant of the Strong Law of Large Numbers

Let $X_1,X_2,\ldots$ be a i.i.d. sequence of random variables with uniform distribution on $[0,1]$, with $X_n: \Omega \to \mathbf{R}$ for each $n$. Question. Is it true that $$ \mathrm{Pr}\left(\...
5
votes
1answer
96 views

Expected value of the shifted inverse of a binomial random variable, and application

Here is an exercise given by a colleague to a student : Let $X\hookrightarrow B(n,p)$ and $Y=\frac{1}{X+1}$. Find ${\rm E}(Y)$. It is not very difficult to prove that the answer is $${\rm E}(Y) = ...
5
votes
1answer
292 views

Sum of 3 uniform random variables is a constant

Give a construction of three random variables $X,Y,Z$ that are each uniform on $(0,1)$ but $X+Y+Z$ is a constant. Is the following argument correct? We first consider every number in $(0,1)$ in ...
5
votes
1answer
129 views

Combinatorial proof that the maximum of an i.i.d. sample of size $n$ uniform on $(0,1)$ has expectation $\frac{n}{n+1}$?

Let $X_i \sim U(0,1)$ for $i = 1,\dots,n$, and let $X = \max(X_1,\dots,X_n)$. Is there a combinatorial way of seeing that $$\mathbb E(X) = \frac{n}{n+1}$$ and likewise for the minimum? Intuitively ...
5
votes
0answers
192 views

How to quantify the “uniformity” of a distribution of holes in a surface

I want to try to quantify if the distribution of holes over a surface is uniform or not. The holes can have any given shape and can be arranged in any way over the surface. Three examples are ...
4
votes
7answers
2k views

Simplest way to produce an even distribution of random values?

I'm a software engineer, working on a small randomizer library as part of a larger project. We're using a cryptographic random number generator, which provides an array of random bytes. We have to ...
4
votes
2answers
1k views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in [-\alpha,\alpha]$, $Y\in [-\alpha,\alpha]$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
4
votes
2answers
9k views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
4
votes
2answers
189 views

Distribution of $\sin(n)$ for $n=1…\infty$

Is the sequence $a_{n}=\sin(n)$ uniformly distributed on the interval $[-1,1]$? For the first $10000$ $n$, it seems as if it is more dense at $-1$ and $1$ than in the middle. Is there any way to ...
4
votes
2answers
101 views

Uniform distribution variable in the Newton symbol

The following question is from actuarial exam. Let $N$ be uniformly distributed on $\{0,1,2,...,19\}$. Compute $$\mathbb{E}\sum_{k=0}^{N}{N-k \choose k}(-1)^k$$ I started $$\mathbb{E}\sum_{k=0}^{N}{N-...