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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Question about a sequence of iid random variables and the Uniform distribution

I will first enuntiate the question and then explain what I'm not understanding. Suppose $ X_1, X_2,\ldots, X_n $ iid with common distribution $ U(0,\theta)$. Define $M$ as follows: $ M : =\max\{ ...
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43 views

Is it possible to generate a uniformly distributed random 128-bit number from multiple uniformly distributed random numbers of size <= 32 bits?

If I have a uniformly distributed random number generator of up to 32 bits in length, can I generate a uniformly distributed 128 bit number by rolling my 32-bit random number generator multiple times ...
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31 views

Location, scale and location scale family

Let $X_1,...,X_n$ be an i.i.d. random sample from a continuous uniform distribution over [0,$\theta$], where $\theta>0$ is a unknown parameter. Does these random variables belong to a family of ...
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1answer
58 views

Roll a die. Are the chances of getting two same consecutive numbers the same as getting any specific random sequence?

Let's imagine a die roll. You roll the die n times. Example: I have a $6$ sided die. Assuming the distribution of the die is perfect, the chances of getting any single number are $1/6$. The ...
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1answer
88 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
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33 views

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,...,n$ Determine the limit ...
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19 views

Standard Uniform Distibution with Random Variable

Could someone help explain how to solve the following problem: From my understanding, this problem states that we have a function, Uniform(0, 1), that will generate a random value from 0 to 1 with ...
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31 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
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48 views

Uniformly at random polynomial

We have a polynomial of degree $d$, and multiply it by a polynomial whose coefficients are chosen uniformly at random and its degree is equal to or less than $d$. My question is whether the result is ...
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1answer
53 views

Exponential random variable

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential r.v. with parameter 1/20. Smith has a used car that he claims ...
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32 views

Average distance between consecutive uniformly distributed values

Say I have a list of values $(X_1, X_2, ..., X_n)$ and $X_i$ is a uniform random variable between 0 and 1. Let $(Y_1,Y_2,...,Y_n)$ be the ordered list of those values. How do I find the expected ...
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122 views

Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
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49 views

Uniform Random Number

Two uniform random numbers are chosen one after the other. what is the probability of second number second random number greater than first number? I tried this way Please correct me if I am wrong. ...
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21 views

Largest hole in uniform sampling of $m$-torus

Let $M$ be the flat m-dimensional torus $(\mathbb R/\mathbb Z)^m$ with the standard Riemannian metric. I would like to know the probability that, given a uniform sampling $X$ of size $N$, there is a ...
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2answers
101 views

Uniform distribution with random support

I have a uniform distribution $ X \sim U(A,B) $ where the limits themselves are random: $A \sim N(\mu_A,\sigma_A^2)$ and $B \sim N(\mu_B,\sigma_B^2)$. Hence the support of $X$ is random. $A$, $B$ are ...
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1answer
92 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 ...
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1answer
34 views

Say (X,Y) has the distribution on the area shown below find P(X>1|Y=1/2) [closed]

Say (X,Y) has the distribution on the area shown below, find P(X>1|Y=1/2)![enter image description here][1]
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108 views

uniform distribution over disk

Given two independent random variables $A$ uniform on $[0,1]$ and $B$ uniform on $[0,2\pi]$. Obtain the joint pdf, tranform to the disk, if necessary modify to obtain the uniform pdf over the disk. ...
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29 views

Let $X$ be uniformly distributed on $[0,1]$. Find the cumulative distribution function of $X-X^2$.

Let $X$ be a continuous random variable uniformly distributed on $[0,1]$. Find the cumulative distribution function of $X-X^2$. $P(X-X^2 \leq a)= P ( -X^2 + X - a \leq 0) = P ( -X^2 + X - a \leq 0| ...
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2answers
71 views

Let U and V be independent continuous random variables, identically distributed uniformly over [0,1]

Let $U$ and $V$ be independent continuous random variables, identically distributed uniformly over $[0,1]$. Show that for $0 \leq x\leq1$ , $$P(x < V < U^2)= \frac{1}{3} - x + \frac{2}{3} ...
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2answers
47 views

Uniform distribution in (0,1). P(X1+X2<=X3) and Gaussian RV with variance 1/4 and 1/9 , P(3V>=2U)

I'm appearing for a competitive examination and I find a lot of questions from probability involving $2$ or more random variables are very common. Please help me with the method on how to deal with ...
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1answer
28 views

Question on distributing weight

My question is about distributing a set of non-negative weights over a set of n items, in a way that sum of weights equals 1. For example if n=2, then w1 can be some p (where p is the probability of ...
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11 views

How can I describe uniformity in a group of values?

I have a matrix of 128x128 (6384 numerical values) referring to light intensity. I will like to know how to calculate the uniformity of the values and I am currently using the standard deviation to ...
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3answers
36 views

$X$ and $Y$ are independent and follow $U(0,1)$. Show $P(f(X) > Y) = \int_0^1 f(x) dx$

Let $X$ and $Y$ be two independent uniformly distributed r.v. on $[0,1]$, and $f$ is a continuous function from $[0,1]$ to $[0,1]$. Show that $P(f(X) > Y) = \int_0^1 f(x) dx$. I tried to prove ...
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131 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
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33 views

The probability that uniformly distributed integers sum to a given integer

A recent CTF had a problem involving the summation of randomly distributed integers. Specifically: Consider a set $\{X_m\}$ of $M$ integers uniformly selected (with replacement) from the set of ...
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1answer
41 views

Discrete Uniform Distribution SOA Practice Problem

X has a discrete uniform distribution on the integers 0,1,2,...n and Y has a discrete uniform distribution on the integers 1,2,3,...n. Find Var[X] -Var[Y] the answer in the book is $ ...
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1answer
36 views

Let X be a random variable with PDF fx. Find the PDF of the random variable |X| in the following

Here's my question: X is uniformly distributed in the interval $[-1,2]$. Find pdf of $|X|$... So I did P($|X| \le x$) = P($-x \le X \le x$)... From here I'm not too sure how to proceed. I know the ...
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36 views

Relationship between quotient of sum of exponentials and uniform distributions

Let $X$, $Y$ and $Z$ be iid with $P(X>t)=e^{-t}$ for $t>0$. Let $U$, $V$ be independent uniform on $[0,1]$. Let $A=\min(U,V)$ and $B=\max(U,V)$. Show that $(A,B),$ and $(X/(X+Y+Z), ...
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81 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 ...
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1answer
81 views

$X$ ~ $uniform(0,1)$, $f_Y (y | X=x) = I( x<y<x+1 )$ ( for $0<x<1$ )

$X$ ~ $uniform(0,1)$, $f_Y (y | X=x) = I( x<y<x+1 )$ ( for $0<x<1$ ). Find.... a) What is the distribution of $Y$, given $X = x$? b) What is $f(x,y)$? Distribution of $(X,Y)$? c) $f_Y ...
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70 views

what is the conditional probability $P(X+Y=2|X-Y=0)$?

Consider two independent random variables $X$ and $Y$ with identical distributions The variables takes values $0,1, 2$ with probabilities $\frac12,\frac14,\frac14$. what is the conditional ...
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1answer
49 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 ...
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1answer
67 views

Expected value of the floor function of a sum of two variables

In a recently published paper I have encountered the following equality. Let $U$ be a random variable uniformly distributed in $[0,1]$ and let $Z$ be a Gaussian variable with mean zero and standard ...
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1answer
68 views

The pdf of $X+Y$

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$) I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But ...
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58 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
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1answer
25 views

What is the probability density function and cumulative distribution function of x

What is the probaiblity density function and cumulative distribution function of $x$ (where $x\in [\dfrac{-\pi}{2},\dfrac{\pi}{2}]$) such that both $y_1=\sin x$ and $y_2=\cos x$ are uniformly ...
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1answer
86 views

Probability question on uniform distribution

I need help with the following question: A computer in adding numbers rounds each number to its nearest integer. Suppose that all rounding errors are independent and uniformly distributed over ...
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35 views

Probability of a certain result obtaioned by throwing an octahedron

Assume having a fair octahedron. We throw it $93$ times and get the following results: $\{33;7;8;1;2;0;5;37\}$ The numbers represent how many times the die fell on side $1, 2,...., 8$. What is the ...
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2answers
120 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 ...
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135 views

pdf of an uniform distribution in matlab?

I'm reading a book and I came across a problem in which I should generate a uniform random variable and use hist, mean and std ...
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23 views

Uniform Spinner is spun twice..

A fair uniform spinner is spun twice, and the results V and W are noted. V and W are uniform RVs ∼U[0,1]. I'm trying to answer the question what is the joint pdf for V and W. I know that I have to ...
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1answer
40 views

Can I uniformly sample from $N$ distinct elements, where $N$ is unknown but finite?

I have access to a list of $N$ elements, but the value of $N$ is unknown. The elements arrive one by one, and never repeat. I want to sample $n$ of these elements as uniformly as possible, as I have ...
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1answer
60 views

$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal

I am looking for a nice proof of the following statement: If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then $$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad ...
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1answer
44 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
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1answer
38 views

What is the probability $P\{X_1 \rm{~is ~largest}\}$? [closed]

Let $X_1,X_2,X_3$ be three independent and mutually identically distributed random variabe with uniform distribution on [0,1]. What is the probability $P\{X_1 \rm{~is ~largest}\}$?
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$X,Y\sim U(0,1)$ 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 ...
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2answers
68 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, ...
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20 views

Injective uniform distribution on an n-sphere

I just asked this question on the stats stackexchange, but I thought that maybe someone on math knew the answer. So: For an application I'm working on, I need to go from some uniformly distributed ...
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30 views

convolution of 3 uniform random variables

Please help. X,Y,Z are uniformly distributed random variates over the closed interval [3,5] independently/ The sum, S, of any of the two random variates X,Y,Z has a triangular distribution with pdf: ...