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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8
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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?
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1answer
124 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. ...
3
votes
1answer
316 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?
1
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1answer
24 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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1answer
75 views

Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $ F(x) = P[X\le x] $ is continuous in x, show that $ Y=F(X) $ is measurable and that $Y$ has a uniform distribution $ P[Y\le y] = y, \; 0\le y \le 1 $ My first question is about notation. What ...
0
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1answer
903 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, ...
17
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0answers
428 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{...
4
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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
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 $(a\in\mathbb{R})\lt (b\in\mathbb{R})$, ...
8
votes
3answers
261 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 ...
5
votes
1answer
356 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 ...
2
votes
2answers
333 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\{\...
0
votes
0answers
55 views

find limit distribution by using central limit theorem.

$$x_1,...,x_n \sim \text{uniform (0,1)}$$ $$Y_n=\sum_i^n X_i$$ I want to find limit distribution by using central limit theorem. $E(Y_n)=n/2$ and $V(Y_n)=n/12$ And Moment generating function $M_{...
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!
5
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
1answer
528 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 ...
3
votes
3answers
762 views

Expected number of overlaps between intervals

Suppose $N$ intervals of length $\delta$ are positioned in $[0,1]$. The starting point $l_i$ of each interval is drawn from an uniform distribution, i.e., $l_i \in [0, 1-\delta]$, thus it will ...
2
votes
2answers
83 views

Maximum of martingales

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, $...
2
votes
1answer
42 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in $\...
2
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2answers
2k 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
1
vote
1answer
1k views

Rao-Blackwell Uniform Distribution

I am having a bit of an argument with my study group about a Rao-Blackwell problem that we have for our statistical theory class. The problem goes like this: Let X~U(0,$\theta$), and suppose we have ...
1
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1answer
935 views

Uniform distribution, Expected value and standard deviation for proportion of observations in a subintervall

$X\sim U(0,1)$. Divide the interval [0,1] into k equal subintervals. Then $X_1$=the number of observtions on the first interval. Define the new variable $Y_1=X_1/n$, where n is the number of ...
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2answers
1k 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 ...
1
vote
2answers
303 views

Distribution of ratio of uniform and exponential random variables

This is a homework question, I feel like I'm doing it right, but I can't seem to get the answer to match up. I have a uniform RV from 2 to 4, and an exponential with mean 4, so $X \sim \text{UNI}(2,4)...
1
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2answers
138 views

Approximate distribution for the sample mean?

A random variable $X$ is said to follow a discrete uniform distribution if its probability function is given by $$p_X(x) = \left\{ \begin{array}{ll}\frac{1}{\theta}, & x = 1, 2, \ldots, \theta\\ ...
1
vote
2answers
69 views

Random variables and Linearity

I have an equation $Y = 5 + 3\times X$ and I assume that $X$ is a random variable taking values from a uniform distribution. Can I consider that also $Y$ is a random variable which takes values from a ...
1
vote
1answer
36 views

Number of uniform hash functions

how many uniform hash function I can create when I want to hash elements from $U$ where $|U|=m \cdot r$ , $m,r$ are integers. a hash function $h:U \rightarrow T $ , $|T|=n$ is uniform if $Prob(h(x)=i)...
0
votes
3answers
53 views

Find the joint distribution of the continuous order statistics?

Take $n$ independent variables, ${X_1, X_2,\dots, X_n}$, which are uniformly distributed over the interval $(0,1)$. Then, introduce the variable $M=\min(X_1, X_2,\dots, X_n)$ and the variable $N=\max(...
0
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1answer
27 views

Finding a uniform distribution on the output of a multivariable function

Suppose we have a non-invertible continuous function that maps from some continuous interval ${I}^n$ to $\mathbb{R}$ with $n \ge 1$. To take an example, let $f(a,b,c) = a \cdot e^{-bc} - b \cdot e^{\...
0
votes
1answer
223 views

Distribution probability of elements and pair-wise differences in a sorted list

Suppose a set of $m$ integers from $0$ to $n-1$. The integers are uniformly distributed and unique in the set ($n \gg m$). Then, put all the integers into a list an sort that list: $$x_0 < x_1 < ...
0
votes
1answer
771 views

Sufficient statistic for uniform distribution

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$. Let ${Y}_{i}$ be the order statistics. Then the sufficient statistic for $\theta$ is ${ Y }_{ n }...
0
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1answer
59 views

Statistics find E(x), the number of distinct elements in uniformly distributed pool of items

Question: Suppose there are Y types of balls in a bucket, which are normally distributed and independent. Hence the probability of picking one type out is $\frac{1}{Y}$. Let $x$ be the number of ...
0
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1answer
50 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\|...
0
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1answer
50 views

Proving that a statistic is not sufficient (uniform case).

Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
-5
votes
2answers
161 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...