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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16
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284 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}^...
10
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538 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. ...
7
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608 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
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91 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
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214 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 ...
5
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0answers
214 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
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52 views

Conditions on angles between three points on a sphere (which are uniformly distributed)

Question: Let $A,B,C$ be three random, uniformly distributed, independant points on a sphere. What is the probability that none of these three points is at an angle superior than $\pi/2$ from the two ...
4
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98 views

Euclidean distance for points in $\mathbb R^2$

I have a point, call it $x$, located somewhere in a unit square $[0,1]^2$. I drawn $n$ new points, all uniformly and independently, all those in the unit square $[0,1]^2$. What is the expected ...
4
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0answers
116 views

Why does this algorithm give the Beta distribution in the limit (considering length of intervals between two random variables)?

Consider the following algorithm: In the $n$th iteration, take two random variables uniform on $[0,1]$. Define the smaller as $X_1^{(n)},$ the bigger as $X_2^{(n)}$ and the interval between them ...
4
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0answers
100 views

$E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$

I am preparing for a test and am unsure of my workings on this exercise: Calculate $E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$ Where $A^+ = \max (A,0)$ My approach was to calculate ...
4
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0answers
2k views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P\left(F^{-1}(F(X))...
3
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22 views

Expected value of general uniform order statistics

I know that if $X_k$ ~ $Unif(0,1)$ and is order statistics, then $E[X_k] = \frac{k}{n+1}$. What's $E[X_k]$ for when $X_k$ ~ $Unif(a,b)$? I think it's $a + \frac{k}{n+1}(b-a).$ Can someone confirm?
3
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273 views

Probability that Carl has to wait at least 10 minutes for one of the others and for both of the others to show up?

This question is for the other subquestions for the same problem here. For those not willing to click the link, I will post the exercise problem here as well. Alice, Bob, and Carl arrange to meet ...
3
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0answers
36 views

Stochastic Independence of $\tan(U_1)$ and $\tan(U_1+U_2)$ for uniform independent $U_1,U_2$

I found the following statement in Stoyanov's book on counterexamples (Section 7.2) quite interesting: Let $U_1$ and $U_2$ be independent and uniformly distributed on $(0,\pi)$. Then $\tan(U_1)$ and $\...
3
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39 views

Finding conditional expectation given another conditional distirbution

Suppose $X$ is a uniformly distributed random variable on $[0,1]$ and given $X=x,$ a number $Y$ is chosen at random between $0$ and $x$. Suppose that you only know the value $y$ of $Y$ and you don't ...
3
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0answers
102 views

Distribution of $ZX+(1-Z)Y$ where $X,Y\sim\mathcal N(0,1)$ and $Z\sim\mathcal U(0,1)$ are independent

Let $X$ and $Y$ be independent $\mathcal N(0,1)$ random variables. Let $Z\sim\mathcal U(0,1)$ be independent of $X$ and $Y$. What is the distribution of $U=ZX+(1-Z)Y$? Clearly, $[U\mid Z=z]=zX+(1-z)Y\...
3
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0answers
185 views

How to find CDF of $|X-Y|$ when X and Y uniformly distributed in a coordinate triangle?

Let $XY$ be two uniformly distributed random variables indicating a coordinate on a triangle $T ∶ ((0,1), (1,0), (0, −1))$, how to find both cumulative and probability density function of $|X − Y|$. ...
3
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0answers
102 views

Probability of random variables in uniform distribution

Suppose I have 2 random variables x1 and x2/2. Where both the random variables lie in a uniform distribution between [0, 1].(that means x1 value lies between [0, 1], whereas x2 value lies between [0, ...
3
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0answers
129 views

Expected root of quadratic random polynomial

Suppose $A,B,C$ are i.i.d. random variables with uniform distribution on $[-1,1]$. I'm interested in the expected roots of the polynomial $Ax^2 + Bx + C$, which are complex random variables given by $$...
3
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0answers
410 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge 0)\...
3
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1answer
89 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...
3
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0answers
62 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
3
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0answers
323 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $\text{Uniform}(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
3
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0answers
322 views

Question on uniform distribution of points on a sphere.

Let N points be uniformly distributed on the surface of a unit sphere $S^2$. What is the probability that every spherical cap of area A contains at least one point? The area $A$ depending on the ...
3
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0answers
120 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all $...
3
votes
0answers
817 views

Expected Value - Uniform distribution over infinite interval

Question: The probability that an error is introduced into a packet is $\alpha$. Messages, consisting of one or more packets, are received at a node. Given that a message has been received free of ...
3
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0answers
56 views

What is the difference between these two questions?

Consider two independent random variables: X~U(0,1) and Y~U(0,2). Let Z = min(X,Y). b) Find F$_Z$(z) in terms of F$_X$(.) and F$_Y$(.). c) Eliminate F$_X$(.) and F$_Y$(.) to find F$_Z$(z). What is ...
3
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1answer
4k views

Derivation of Variance of Discrete Uniform Distribution over custom interval

I'm trying to prove that the variance of a discrete uniform distribution is equal to $\cfrac{(b-a+1)^2-1}{12}$. I've looked at other proofs, and it makes sense to me that in the case where the ...
3
votes
2answers
1k views

Pdf for distance between two uniform random points in a circle

This is my first post in the group and I would be very thankful for any help. I am trying to develop a probability distribution for a performance analysis in my thesis. I am trying to look in to ...
2
votes
2answers
153 views

Uniform Random Variable on $[0,1]$ and Bernoulli$(1/2)$

Let $X_1,X_2,...$ be independent, identically distributed (iid) random variables with distribution Bernoulli$(1/2)$. Define the random variable: $$Y=\sum_{n=1}^\infty\frac{X_n}{2^n}.$$ Then $Y$ is ...
2
votes
1answer
21 views

showing the pdf of n-th order statistics

I am working on a mathematical stats assignment and I got stuck here. Letting $X_1, X_2, ... ,X_n$ a random sample from uniform(0,$\theta$), and Y is n-th order statistic, I need to show that the ...
2
votes
1answer
44 views

How to compute a conditional expectation

I want to compute a conditionnal expectation, i know that $Z=(Z_1,\ldots,Z_p)'$ where $ Z_j=\Phi ^{-1}(U_j)$ with $Z \sim N(0,R(\theta))$ and $R(\theta)$ the $p \times p$ positive definite ...
2
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0answers
78 views

Variance of sum of two uniform RV

Let $X$ and $Y$ be two independent random variables, each uniformly distributed on $[-1,1],$ then find $\operatorname{Var}(X+Y).$ My attempt : $$\operatorname{Var}(X+Y) =\operatorname{Var}(X) + \...
2
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0answers
98 views

Show that $P(N \geq n) = (1-e^{-\lambda})^n/ \lambda^n $

The lifetime $X$ (in days) of a device has an exponential distribution with parameter $\lambda.$ Moreover, the fraction of time which the device is used each day has a uniform distribution over the ...
2
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0answers
80 views

“Fragmentation” of a distribution (from paper)

I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to ...
2
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0answers
60 views

Normal approximation of sum of uniform independent RVs using CLT

Let $X_1$, $X_2$, ... $X_{16}$ and $Y_1$, $Y_2$, ... $Y_{16}$ be independent uniform random variables over the interval [-1,1] and let: $$ W = \frac{(X_1 + .... + X_{16}) + (Y_1 + .... + Y_{16})}{16} ...
2
votes
2answers
38 views

On the probability of forming a triangle, when $(0,1)$ is divided into three segments , where dividing points are i.i.d. Uniform $(0,1)$

Divide $(0,1)$ into three line segments, let $X,Y$ be the dividing points. Assume $X,Y$ are independent and follows Uniform $(0,1)$. What is the probability that the three line segments can form a ...
2
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0answers
208 views

How can I calculate the joint probability for three variable?

I am a student studying the joint probability density function with multi variables. I understand how to obtain a joint probability density function when two uniform distributions have the following ...
2
votes
1answer
120 views

calculate marginal PDF from joint PDF of dependent random variables

The marginal PDF $f_X(x)$ can be calculated as $$f_X(x)=\int f_{X,Y}(x,y)dy=\int f_{X|Y}(x|y)f_Y(y)dy \tag{1}$$ However, I stuck in a particular case as follows. $\mathbf{X}=[X_1,X_2]$ is uniform ...
2
votes
1answer
24 views

Independent variables' probability functions and expectations

So, I've got this exercise Two independent random variables $X,Y$ ~ Uniform $[0, 1]$. Find the probability function of the random variable $Z=X−Y$. Compute expectation of $E [Z]$ So, I need to find ...
2
votes
2answers
28 views

Probability in uniform

I need help to solve the following problems. Thank you in advance. Problem 1: A random variable $X$ is uniform $[0, 1]$. Find the probability that X's 2nd digit is $3$. As far as I understand it ...
2
votes
0answers
69 views

Probability of being closest to each other

Suppose there are $N$ red dots and $M$ blue dots uniformly distributed in a square region with side $S$. Each red dot finds its closest blue dot and each blue dot finds its closest red dot. What is ...
2
votes
0answers
74 views

Conditional distribution of $Y$ given $X+Y$ when $X \sim$ Unif$(-a,a)$ and $Y \sim F$

THE PROBLEM HEURISTIC SOLUTION Since $X \sim$ Unif$(-a,a)$, the restriction that $X+Y=u_0$ automatically imposes the following restriction on $Y$ : $$Y \in (u_0-a,u_0+a) \tag{1}$$ Hence the ...
2
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0answers
126 views

Convergence of ratio of two sums of uniform random variables

Consider the sequence of rectangles, which sides are length $(X_1, Y_1), (X_2, Y_2),...,$ where $X_1, X_2,...$ have uniform distribution on $(0,2)$ and $Y_1, Y_2, ...$ have uniform distributions on $(...
2
votes
0answers
17 views

integrand of norm subjected to translation

Sorry if my title makes confusion. Let $\mathbf{x} \in \mathbb{R}^n$ is uniformly distributed on a $(n-1)$-sphere of radius $\sqrt{nP}$, thus $\left\Vert \mathbf{x} \right\Vert^2=nP$. Obviously, $$\...
2
votes
0answers
31 views

A problem about effective uniformly distribution

The original problem is following: Problem 1 $\lim_{N\to \infty}|\frac{\#\{1\leq n\leq N |\ \ \ [n\sqrt2]=0(mod 2)\}}{N}|=\frac{N}{2}+O(ln(N))$. This problem is not very difficult, in fact $\#{\{...
2
votes
0answers
97 views

Constructing joint confidence intervals/multiple confidence intervals

Consider $n$ iid observations $X_1,X_2,\dots ,X_n$ from a $Uniform(a,b)$ distribution, where $a$ and $b$ are both unknown. How do we construct a joint confidence interval for $(a,b)$? I would prefer ...
2
votes
0answers
163 views

Transform n-dimensional standard normal data to a uniform distribution on the unit n-sphere

While thinking of some algorithms related to machine learning, I went on a tangent and eventually asked myself if I could transform a standard normal distribution into a uniform distribution on the ...
2
votes
0answers
567 views

What distribution to use for prior if likelihood is uniform?

I am trying to find the posterior predictive distribution of a future test case of a Bayesian model, but I'm stuck on which prior to use, and how to integrate it with the likelihood to obtain the ...
2
votes
0answers
365 views

Why isn't the Laplace-Stieltjes transform of the uniform distribution at $s = 0$ equal to 1?

I've been studying the Laplace-Stieltjes transform and I noticed that, according to the definition, the Laplace transform of any probability distribution at point $s = 0$ should be 1. The transform ...