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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1answer
34 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 ...
-1
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0answers
12 views

what are the joint distribution functions and copula? [closed]

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ A &=&\frac{U+V}{2}, \\ G &=&\sqrt{UV}, \\ H ...
1
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1answer
55 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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0answers
50 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 ...
0
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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
38 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
29 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 ...
4
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2answers
88 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
30 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 ...
0
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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 ...
1
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1answer
32 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 ...
2
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1answer
29 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 ...
0
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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
32 views

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

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 $P\{X_1 \rm{~is ~largest}\}$?
2
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0answers
39 views

$X,Y\sim U(0,1)$ 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
47 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
18 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 ...
0
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0answers
10 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: ...
0
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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
168 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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0answers
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 $Q=\max\{X_i,i=1,2,3..,n\}$. ...
2
votes
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 ...
4
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3answers
49 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]. ...
1
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1answer
72 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
60 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 ...
0
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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 ...
1
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1answer
32 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
39 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
7 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 ...
0
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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 ...
2
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2answers
38 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 ...
2
votes
1answer
37 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 ...
3
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1answer
84 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 ...
0
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1answer
27 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 ...
0
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0answers
14 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
17 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 ...
2
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1answer
107 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 ...
1
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1answer
22 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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0answers
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
31 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 ...
2
votes
1answer
43 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 ...
0
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0answers
33 views

Expectation and Variance of a Discrete Uniform Distribution using the Probability Generating Function and Cumulant Generating Function

Hi I just derived the MGF of a discrete uniform distribution and found it to be: [e^t - e^t(m+1)]/(1 - e^t)m and the pgf is ...
2
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0answers
38 views

Expected value of winning with a Random Uniform Variable

I have this small homework I don't quite understand. The problem is formulated as a game. (Who wants to be millionaire) You start with 1£. money = 1.0£ You can choose to quit at anytime So you can ...
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1answer
58 views

Expected value of multiple random variables, uniform distribution

Suppose that the random variables $X_1,\dotsc,X_n$ form a random sample of size $n$ from the uniform distribution on the interval $\left[0, 1\right]$. Let $Y_1 = \min\left\{X_1,\dotsc,X_n\right\}$, ...
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1answer
51 views

Power Function for the uniform distribution

Completely stuck on this homework question, I think my knowledge of the power function is nowhere near good enough coming up to finals! Consider the following alternative testing problem: the two ...
8
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3answers
188 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 ...
2
votes
1answer
30 views

Verify conditions of the Extension Theorem for a probability measure $\mathbb P$ of the Uniform[0,1] distribution

I am (self-)studying the book by Rosenthal called A first look at rigorous probability theory. My question is on verifying the conditions on a probability measure $\mathbb P$ of the Uniform[0,1] ...
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1answer
44 views

Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...