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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15
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0answers
409 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 ...
8
votes
3answers
259 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
0answers
210 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
2answers
7k 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?
6
votes
2answers
2k 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
1answer
2k 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!
6
votes
0answers
64 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 ...
5
votes
2answers
107 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
3answers
2k views

Binomial distribution with Uniform parameter

I have a problem with following exercise (it comes from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001, page 155, ex. 6): Let $X$ have the ...
5
votes
1answer
56 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 ...
5
votes
2answers
616 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
1answer
354 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
0answers
49 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
2answers
5k views

Distribution of Max(X_i) | Min(X_i), X_i are iid uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) | \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$ I agree that the probability ...
4
votes
2answers
3k 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 $c_1$ ...
4
votes
2answers
370 views

Sum of discrete and continuos 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 ...
4
votes
2answers
697 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?
4
votes
2answers
159 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
4
votes
2answers
634 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 ...
4
votes
3answers
139 views

$P\{B^{2}-4AC\geq 0\}$ where $A,B,C \sim U(0,1)$?

The actual problem is to find the probability that $Ax^{2}+Bx+C=0$ has real roots. This boils down to whether or not the discriminant $B^{2}-4AC$ is non-negative. Thus, we seek $P\{B^{2}-4AC\geq 0\}$. ...
4
votes
1answer
312 views

The “beach problem”: does anyone know it? or know how to solve it?

The following problem was given some years ago in the German computer-science contest for pupils ("Bundeswettbewerb Informatik"). It was originally wrapped in a story which I will briefly translate ...
4
votes
3answers
3k 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]. ...
4
votes
1answer
523 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 ...
4
votes
1answer
240 views

Why do prime numbers in modulo result in more uniform distributions?

Let us assume a sequence as follows: $S_{n} = (S_{n-1} * c_{1} + c_{2})\text{ mod } m$ This is the pseudorandom generator found in most programming languages' random function. It is known that a ...
4
votes
2answers
67 views

Probability problem: length of new segments

I have a line of length $l$. I divide the line in $n$ segments. I do this by choosing $n - 1$ random points (I mean that the $n - 1$ points are uniformly distributed from $0$ to $l$). I want to add a ...
4
votes
0answers
85 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 ...
3
votes
4answers
314 views

Find the probability of $a>b+c$, where $a$, $b$, $c$ are $U(0,1)$

What is the probability that $a > b + c$? $a, b, c$ are picked independently and uniformly at random from bounded interval [0,1] of $\mathbb{R}$.
3
votes
3answers
71 views

Inductively defined random variables

Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$. Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the ...
3
votes
2answers
126 views

Why is $P(X<0)$ the same as $P(X\le 0)$ for continuous distributions?

In uniform distribution, a continuous distribution, for example where $A = -1$ and $B = 1$, $P(X < 0)$ is said to be the same as $P(X \le 0)$. Why is this?
3
votes
1answer
5k 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 ...
3
votes
2answers
38 views

How to show that $y=Px$ is distributed like binary $x$ for random permutation $P$?

Drawing a random binary vector $X\in\{0,1\}^n$ from the uniform distribution, the probability $\mathbb{P}(X=x)$ to get a specific $x\in\{0,1\}^n$ is known ($=\frac{1}{2^n}$). Let ...
3
votes
4answers
107 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 ...
3
votes
2answers
109 views

How to show $\max\{Y_{1},Y_{2},\cdots,Y_{n}\}$ converges in probability to $\theta$ as $n \to \infty$. [closed]

Let $Y_{1},Y_{2},\ldots,Y_{n} $ be independent random variables , each uniformly distributed over the interval $(0,\theta)$. Show that $\max\{Y_{1},Y_{2},\ldots,Y_{n}\}$ converges in probability to ...
3
votes
2answers
2k 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 ...
3
votes
1answer
311 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
votes
1answer
269 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
3
votes
2answers
122 views

A single, good test for a random number generator?

I'm chasing a bug in the RNG for a well-known programming language under certain pathological inputs. There is an obvious pattern in this pathological case, apparent with very small n (~ 10000), and ...
3
votes
2answers
542 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
votes
1answer
31 views

Is modular multiplication under a prime modulus uniformly distributed?

Let's say that I have a prime, $p$, and an $m \in Z_p^*$. Then, I draw $a \leftarrow Z_p^*$ uniformly at random. Will $am \mod p$ be distributed uniformly over $Z_p^*$?
3
votes
2answers
63 views

Pseudorandom Number Generator Using Uniform Random Variable

I am working out of Mathematical Statistics and Data Analysis by John Rice and ran into the following interesting problem I'm having trouble figuring out. Ch 2 (#65) How could random ...
3
votes
1answer
66 views

Collisions with four bullets

This is a follow-up question of Colliding Bullets. I'm interested in a rigorous calculation of a specific aspect of the referred question. We consider four bullets. Once per second a bullet is fired ...
3
votes
1answer
64 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
3
votes
1answer
57 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
3
votes
1answer
37 views

If $X$ ~$Uni(-1,1)$ show that $X$ and $X^2$ are not independent

I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$. $F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence ...
3
votes
1answer
40 views

CDF on Standard uniform gives the same distribution

Assume that $X$ has a continuous and strictly increasing CDF $F_X$. Define $Y = F_X^{-1}(U)$ where $U$ is standard Uniform. How dow I show that $X$ and $Y$ have the same distribution?
3
votes
1answer
567 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
3
votes
1answer
33 views

Average waiting times

I have the following exercise, which I would like to solve: Company A run buses between New York and Newark, their bus leaves New York every half an hour starting from 0:00, 24h a day. Company B also ...
3
votes
1answer
33 views

Expected value of a series of random variables

There are $K$ checkout counters in the mall, and there are $N$ shoppers in the queue waiting for a checkout counter. Initially all counters are empty. Whenever a counter is empty, the next shopper in ...
3
votes
1answer
177 views

Given 3 random points, what is the probability of these two situations involving a perpendicular bisector and distances?

Suppose we're given 3 random points $p_0=(x_0,y_0),p_1=(x_1,y_1),p_2=(x_2,y_2)$ from a two-dimensional continuous uniform distribution $\{U(a,b)\}^2$, for some $(a\in\mathbb{R})\lt (b\in\mathbb{R})$, ...
3
votes
1answer
141 views

Circular distribution of circles

Suppose you have $n$ objects , distributed randomly, in a circular manner of radius $r$. Each objects is of area $A$. So my question is if you draw line everywhere from the center to the surface of ...