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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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
21 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
21 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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24 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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2answers
40 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
73 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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2answers
67 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
40 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
41 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
62 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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52 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
17 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
39 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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1answer
34 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
94 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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0answers
38 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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1answer
21 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
34 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
34 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
33 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
37 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
49 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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0answers
19 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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0answers
11 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: ...
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2answers
74 views

What is the pdf of $X,Y$?

We know that the common pdf of $X,Y$ is constant function, on the triangle $(0,0),(0,1),(2,0)$ (and out of this range the value of the function is zero). What is $f_X(x)$ and $f_Y(y)$? My solution: ...
3
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1answer
170 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 ...
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17 views

The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$. Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. ...
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1answer
38 views

Finding the distribution of $M$

Let $(X_1,X_2)$ be uniformly distributed in $[0,1]^2$ and define $Y_1=\max(X_1,X_2)$, $Y_2=\min(X_1,X_2)$. What is then the distribution of $M:=Y_1-Y_2$ ? To find the joint distribution we take a ...
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0answers
23 views

Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
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0answers
10 views

Uniform distribution over an hyper-ellipsoid

Let $\mathbf{X} \in \bf{R}^p$ be a random vector whose elements are uniformly distributed over the hyper-ellipsoid $x^TAx<1$, (where $A$ is a positive-definite matrix). Is it possible to compute ...
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3answers
50 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]. ...
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1answer
74 views

Convolution of two Uniformly distributed r.v. ove

Assume a continuous random variable $X$ that is uniformly distributed $\underline{\text{on}}$ a $k$-sphere. For simplicity, lets assume a simple circle with radius $R$ in 2 dimension. Therefore ...
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3answers
62 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
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1answer
39 views

continuous RV from discrete RV

So I am reading some notes in stochastic processes and I don't really understand the solution of this problem: Problem: Let $(\Omega,F,\mathbb{P})$ be a probability space where $\Omega$ is the set ...
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1answer
36 views

Proving that a moment generating function converges pointwise

I have found a moment generating function $M_n$ given by $\cfrac{(1-e^t)e^{\frac tn}}{n(1-e^{\frac tn})}$ if $t\ne 0$ and 1 if $t =0$ How do I prove that $M_n$ converges point-wise to the moment ...
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0answers
45 views

Second moment of random variable in the integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
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0answers
9 views

PDF describing nth term in continued fraction

For a real number r chosen uniformly at random in the range (0,1), what's the marginal Probability Density Function that describes the nth term in the continued fraction representation of r? What ...
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1answer
15 views

Distribution function of the random variable $R_2=e^{-R_1}$

An absolutely random variable $R_1$ is uniformly distributed betweem $-1$ and $+1$, find the density and the distribution function of the random variable $R_2$, where $R_2=e^{-R_1}$. $R_1$ is ...
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2answers
39 views

Uniform Distribution Problem

Let $X$ be a random variable uniformly distributed in $[0,1]$, and let $Y$ be a RV uniformly distributed in $[X,1]$. I want to calculate the theoretical distribution of $Y$, any hints? I already tried ...
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1answer
42 views

P.d.f. of $XY$, where $X, Y$ are independent uniformly distributed over $[0,1]$ [duplicate]

I tried to change the variables: Let $U=XY$ and $V=Y$; so then the Jacobian is $1/v$. So joint pdf $g(u,v) = f(x,y)\cdot (1/v) = 1/v$ Would you then integrate over $v$ from $0$ to $1$ to get the ...
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1answer
86 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 ...
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1answer
28 views

Finding expected value of E(Y^2)

I have an equation that looks like this: $11.16^2 = X^2 + Y^2 \implies 124.5456 - X^2 = Y^2 \implies 124.5456 - E(X^2) = E(Y^2)$ is that correct? The X is random variable that is distributed by ...
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0answers
17 views

Random uniform distribution using the Jacobian

In polar coordinates $(r, \theta)$, to get a uniformly random point where $r\in[a,b]$ and $\theta\in[\alpha,\beta]$, since the Jacobian is $r$, you would need to first randomly pick $r^2\in[a^2, b^2]$ ...
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1answer
18 views

MLE for lower bound of Uniform Distribution

Let $X_1$, $X_2$, . . . , $X_n$ be a random sample from a $Uniform(θ, 1)$ population, where $θ < 1$. (a) Find the MLE $\widehat{\theta}$ of $θ$. (b) Find constants c and d (possibly depending on ...
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1answer
113 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 ...
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1answer
24 views

Two Uniform Independent Random Variables: When is one greater?

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$. Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality ...
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14 views

Fast uniformity test within a ball.

Assume I have a dataset lies within a ball centered around the origin, I want to test the uniformity of the point distributed in the ball. In addition, I have all the distances to the origin computed ...
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2answers
33 views

Standard Uniform Distribution

I am trying to show that a random variable $X_2$ has a standard uniform distribution. I have: $\alpha \subset(0,1), X_1 \sim U[0,1],$ and $X_2 = \begin{cases} X_1 &\mbox{if } X_1< \alpha ...
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1answer
48 views

Finding correlation of Max and Min of two IID random variable in U[0,1]

I have a hw problem and can't figure out how to do it. Basically, $X,Y$ are iid $U[0,1]$, we need to find the correlation between max$(X,Y)$ and min$(X,Y)$. My thought is to find the pdf of ...