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.

15
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245 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}^...
13
votes
0answers
358 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
0answers
516 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
votes
0answers
555 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
0answers
195 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
88 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
0answers
201 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
0answers
51 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
votes
0answers
95 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
votes
0answers
115 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
votes
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
votes
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
votes
0answers
121 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
34 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
votes
0answers
38 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
votes
0answers
97 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
votes
0answers
161 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
votes
0answers
95 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
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
0answers
118 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
votes
0answers
378 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
votes
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
votes
0answers
298 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
votes
0answers
314 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
votes
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
743 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
votes
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 ...
2
votes
0answers
37 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
votes
0answers
92 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
votes
0answers
74 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
47 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
0answers
187 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
0answers
67 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
70 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
votes
0answers
97 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
29 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
105 views

Uniformly distributed in Poisson-Voronoi cell

When I read research article: here, I found following set-up: Figure illustration of one realization of Big node (BN) positions from the homogeneous Poisson point process (PPP) and of small nodes (SN) ...
2
votes
0answers
77 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
153 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
518 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
310 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 ...
2
votes
0answers
58 views

Probability that collections of draws from discrete uniform contain the support

Consider a finite set $A$ consisting of $n$ distinct elements. Draw $m$ independent elements $a_1, \dots, a_m$ with replacement from $A$. Define $B=\{a_1, \dots, a_m\}$. What is $P(A\subset B)$? ...
2
votes
0answers
76 views

Uniformly distributed subsequence

Let $(x_{n})_{n\geq 1}$ be a sequence of real numbers uniformly distributed mod 1. If $E$ is a subset of $\mathbb{N}$ where the limit $$d(E)=\lim_{N\to\infty}\frac{1}{N}|\{1, 2, \dots, N\}\cap E|$$ ...
2
votes
0answers
27 views

A regression model for the discretization of $U \sim unif[0,1]$.

Let $U \sim unif[0,1]$ and $U_n = \frac{\lfloor nU\rfloor}{n}$. a) Determine the distribution of the difference variable $W_n = U - U_n$. b) Using part a), evaluate the correlation coefficient $\rho(...
2
votes
0answers
435 views

Statistics $X_{(1)}$ complete for a Uniform Distribution?

Someone had asked this earlier, but since it was good practice for my qualifying exam coming up, I figured I would ask and share my work on the problem. The problem is: Suppose $X$ is Unif$(0,\...
2
votes
0answers
63 views

Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\rightarrow 0$ in probability.

This is a qual problem。 Let $n$ points be iid uniformly distributed on the unit circle. Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\...
2
votes
0answers
50 views

Targetting the distribution of distances between points

For a certain problem, I need to make distance dependent statistics, but with the constraint that the number of sampling points, $N$, should be kept as small as possible. To be more specific I need to ...
2
votes
0answers
577 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...
2
votes
0answers
41 views

convergence in distribution

Let $U_{t}$ be iid Uniformly distributed on (0,1). Suppose $\hat{\theta}_{T}\stackrel{d}\rightarrow \theta^{*}$ with $\theta^{*}$ some random variable on (0,1). I believe $\sum_{t=1}^{T}I(U_{t}\leq \...