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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.

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9
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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?
10
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1answer
8k 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!
0
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3answers
3k views

density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution

Suppose $X$ has uniform $[-1,2]$ distribution. I am trying to find the density of $Z=X^2$. Here is what I have done thus far: Range($Z$)$=[0,4]$. I began computing the distribution of $Z$ for $z \in ...
8
votes
2answers
7k 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?
1
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2answers
191 views

Dumb question: Computing expectation without change of variable formula

Possibly related question: Making sense of measure-theoretic definition of random variable Given a random variable $X$ on $(\Omega, \mathscr{F}, \mathbb{P})$, its law $\mathcal{L}_X$ and a Borel ...
55
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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{...
5
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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, \...
9
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1answer
9k 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 ...
10
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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 ...
5
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0answers
215 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 ...
1
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1answer
129 views

How To prove Any Change to $v=a\cdot y + b$ maks $y=(a)^{-1}(v-b)$ Uni. random value

This question may seem to be related to Probability and Data Integrity but mine is much simpler and consideres a DIFFERENT problem. Let a finite field be $\mathbb{Z}_p$, where $p$ is a prime number. ...
1
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1answer
157 views

Probability density function for the radius half the length of two equal intersecting circles

Considering this diagram, assuming a uniform distribution in the area of UQWD, it is still not clear how the probability density function of r becomes $l(r)/S$. Where S is the area of UQWD. What is ...
28
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2answers
31k 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 ?...
13
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2answers
7k 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 ...
5
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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 ...
12
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2answers
19k 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 $c_1$ ...
8
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2answers
146 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) = ...
5
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2answers
6k 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 ...
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2answers
4k views

Probability that sum of independent uniform variables is less than 1

I would like to determine the probability $\mathbb{P}(X_1+\dots+X_n\leq 1)$, where $X=(X_i)_{1\leq i\leq n}$ is a family of independent uniform random variables on $[0,1]$. My first idea is to do this ...
4
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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 ...
3
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1answer
619 views

Show that there is no discrete uniform distribution on N.

This is a homework question I got. I'm not entirely sure what it is asking. Can someone please clarify/get me on the right track?
3
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4answers
3k views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that $X_{(1)}=s_1,X_{(2)}=s_2....,...
1
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1answer
704 views

Why is $X/\|X\|_2$ uniformly distributed on a unit sphere when X is n-dimensional standard gaussian vector?

In the proving the above, I see that since $X$ is multivariate gaussian then for any orthogonal matrix $Q$ we have that $QX$ is standard multivariate gaussian. Then I somehow reasoned that $Y=X/\|X\|...
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1answer
2k views

equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
0
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1answer
110 views

Uniform Distribution - Finding probability distribution of a random variable

Given the random variable X has a uniform distribution $$f(x) = 1$$ $$\text{if }1<x<2$$ Find the probability distribution of the random variable: $$Y = -2\ln x$$ Could you anyone please ...
2
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0answers
120 views

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$.

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$. How do I find the PDF of $W$? How do I find the expectation of $W$ at two ways: 1. with the PDF of $W$ and without the PDF of $W$. I ask this Q ...
2
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1answer
47 views

Why $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent if $X_1$ and $X_2$ are i.i.d. exponential?

How to show that $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent, if $X_1$ and $X_2$ are i.i.d. exponential with mean $1$? Is there a simple way to see this?
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0answers
56 views

Q: joint density $P(x, y)$ when $Y=X^2+Z$ and $X, Z$ are uniformly distributed.

$Y=X^2+Z$, $X$ and $Z$ are uniformly distributed. $$ P(x)=\frac12 I_{\{-1<x<1\}}, \\ P(z)=10 I_{\{0<z<1/10\}}\\ P(x,y)={}? $$ How can I solve it? **X and Z are independent
1
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1answer
232 views

Prove effiency of this discrete uniform distribution estimator

I have a sample of $n$ random variables with discrete uniform distribution over $\{1, \dots, \theta\}$. I have to prove that $$T = \frac{X_{(n)}^{n+1} - (X_{(n)} - 1)^{n+1}}{X_{(n)}^{n} - (X_{(n)} - 1)...
0
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1answer
4k views

Finding the distribution of the sum of three independent uniform random variables

Let $X,Y,Z\sim Unif(0,1)$, all independent. Find the distribution of $W=X+Y+Z$ I'm trying to solve this by doing convolution twice. I'm letting $S=X+Y$, then $W=S+Z$. So I should end up with $f_W(w) ...
16
votes
1answer
435 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 ...
9
votes
1answer
350 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 ...
21
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2answers
607 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 ...
7
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2answers
5k 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 ...
5
votes
2answers
83 views

Expected number of regions with $n$ random lines in a circle

There are $n$ random lines drawn in a circle, defined by endpoints being uniform on circle. I am trying to figure out the expected number of regions separated by $n$ lines. I know $f(0)=1$, $f(1)=2$ ...
3
votes
4answers
11k views

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece
2
votes
2answers
1k views

Random Walk on Clock Hands

We do a random walk on a clock. Each step the hour hand moves clockwise or counterclockwise each with probability 1/2 independently of previous steps. If you start at 1 what is the expected number ...
4
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1answer
1k views

Rao-Blackwell's Theorem for uniform distribution

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[\theta-\frac{1}{2}, \theta+\frac{1}{2}]$ with $\theta \in \mathbb{R}$ unknown. Take for granted that $T(\mathbf{X}) = (X_{(1)}, X_{(n)}...
5
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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
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3answers
20k 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]. ...
3
votes
2answers
773 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process $\left\{\...
1
vote
1answer
8k views

Probability density of Continuous uniform distribution over the unit circle

If we want to chose a point $(x,y)$ uniformly at random from a unit circle in a plane, why is the joint probability density of the random variable $f(x,y) = \frac{1}{\pi}$ for $x^2+y^2\leq1$? The ...
1
vote
3answers
957 views

PDF of sum of random variables (with uniform distribution)

How can I solve this: Random variables $X,Y$ ~ Unif$(0, 1)$ are independent. Calculate the probability density function of sum $X + 3Y$. I couldn't find a sum for uniformally distributed random ...
0
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3answers
507 views

Independence and Order-Statistics [closed]

Let $X_1,...,X_n$ be a random sample from a uniform $U(0,\theta)$ distribution, where $\theta>0$. Are $X_{(n)}$ and $\left(\frac{X_{(n)}}{X_{(n-1)}},\frac{X_{(n-1)}}{X_{(n-2)}},...,\frac{X_{(2)}}{...
8
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3answers
327 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 ...
7
votes
1answer
388 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
1answer
406 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 ...
5
votes
2answers
11k 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 ...
5
votes
2answers
1k 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 ...
4
votes
1answer
2k views

Is there a concept of asymptotically independent random variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...