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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3answers
49 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 ...
1
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2answers
56 views

Compute $\mathbb{E}(U_{1} | M)$ and $\mathbb{P}(U_{1}=M)$ [closed]

I am having trouble getting started on and finishing this problem. Any help that can be offered would be greatly appreciated. Let $U_{1},...,U_{n}$ be independent random variables uniformly ...
1
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1answer
29 views

Probability that a sample represents between X% and Y% of the population

Really no idea how to go about this. I thought about using a uniform normal distribution law but the answers I got made no sense. In a country that has a population between 1500000 and 3000000 ...
1
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1answer
48 views

RVs independently and uniformly distributed on interval $[0,1]$, prove every order is equally likely

Finitely many random variables $p_1,p_2,...,p_n$ are independently and uniformly distributed on interval $[0,1]$. They form an ascending sequence $p_{i_1} \le p_{i_2} \le ... \le p_{i_n}$. For ...
0
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2answers
69 views

The distribution of the x-coordinate on unit circle

I'm trying to determine the distribution of the x-coordinate (uniformly distributed) on the unit circle (density function). I've seen some threads on this, such as this, where they use the method of ...
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0answers
24 views

Could you please explain how do we get the second term while computing $E[f(Y)]$

I will fix the dimension $n$ and use $S:=\{x\in\Bbb R^n:\|x\|=1\}$ to denote the unit sphere. Let $\sigma$ denote surface measure on $S$, and define $\bar\sigma:=[\sigma(S)]^{-1}\sigma$, the "uniform ...
0
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1answer
27 views

Sum of transformations of continuous uniform random variable

Let $X$ be uniformly distributed on $(a,b)$. I want to find the cdf of $$ \sin^2(X) + \cos^2(X) $$ My feeling is that since $\sin^2(X) + \cos^2(X) = 1$, the cdf will be: $$F(1 \le x)= \begin{cases} ...
1
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3answers
114 views

Find the probability of $X_2$ lying between $X_1$ and $X_3$

All $X_1,X_2,X_3$ are independent and uniformly distributed on $[0,1]$. Find the probability of $X_2$ lying between $X_1$ and $X_3$ Is the following method correct? Find the ...
2
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2answers
46 views

Probability of getting the same number three times.

If I have a set of numbers $\{1 \ldots n\}$ where $n \ge 1$ and I pick $3$ numbers from the set independently and uniformly. Whats the probability I'll get all $2$'s, the probability I get all the ...
0
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2answers
51 views

Find the probability of $\max[X,Y]<\frac{1}{2}$?

$X$ and $Y$ are uniformly distributed random variable on $[-1,1]$. Find the probability of $\max[X,Y]<\frac{1}{2}$. I calculated it using graphically that the region where the where $[X,Y]$ is ...
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0answers
32 views

Uniform distribution density function

Let's say we have two random variables $T_1$ and $T_2$ and the joint density function of the two is uniform over the region $0\leq t_1\leq t_2 \leq L$, where L is a positive constant. Then the area of ...
1
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1answer
36 views

question about the conditional disjoint probability.

I am trying to solve a problem like the following. Q) The event that a man arrives at a bank $\sim Poisson(\lambda)$. If two men visited the bank between 9:30 AM and 10:30 AM, what is the ...
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0answers
17 views

Find probability coverage under a discrete uniform

Let $Y_1=\theta+\epsilon_1$ and $Y_2=\theta+\epsilon_2$ where $\epsilon_1$ and $\epsilon_2$ are iid uniform in {-1,1}. Define the confidence set S for $\theta$ as: $S={Y_1-1}$ if $Y_1=Y_2$ and ...
0
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1answer
28 views

Math Behind Producing Uniform Distribution

I am familiar that if one can produce a uniform distribution, doing so, one can then produce random numbers for other types of distributions. I have tried reading some articles online but I am still ...
2
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1answer
95 views

All Cdfs have a uniform distribution on [0, 1]?

Consider the following proposition Proposition C Let $Z = F(X)$; where $F$ is the continuous cumulative distribution function of the random variable X, then $Z$ has a uniform distribution on ...
0
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1answer
49 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 ...
1
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1answer
75 views

Uniform distribution over the unit disk

Suppose that $U_1$ and $U_2$ are independent, and identically and uniformly distributed over the unit disk, i.e., for $i = 1,2$, $U_i = (X_i, Y_i)$ and the joint density is \begin{equation} ...
3
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2answers
66 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 ...
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0answers
54 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 ...
1
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1answer
34 views

Distribution of a Poisson process with uniformly random parameter

Let $X = Unif(2, 4)$ and $Y=Poisson(X)$. My goal is to find $P(Y=n)$, but I always seem to get stuck on some nasty integral. Here's what I've tried: $P(Y=n) =\int_2^4P(Y=n|X=x)*P(X=x)dx = \int_2^4 ...
4
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2answers
71 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 ...
2
votes
1answer
60 views

Two step random experiment: Density of combined uniform and normal distribution

imagine a random experiment, where first some number $u$ is drawn uniformly on $[c-\varepsilon,c+\varepsilon]$ for $c>0$ and $0<\varepsilon<c$. Next, a $N(u,\sigma^2)$-distributed random ...
0
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2answers
34 views

Sum of X-Uniform(0,1) + Y-Uniform(0,2)

I'm trying to find the CDF the sum of $X$ and $Y$ (which are independent). $X$ is uniform distributed over $[0,1]$ and $Y$ over $[0,2]$. I've seen some similar questions which explain the situations ...
0
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3answers
62 views

Generate prime of x decimal digits using bit-oriented prime generator

I've got a question on stackoverflow where somebody asks to generate a random 18-digit prime. Unfortunately, the only prime generator is the one from OpenSSL. This prime generator is however geared ...
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1answer
54 views

How do I renormalize these probability distributions?

So I have two random variables, $X_1$ and $X_2$, both uniformly distributed on $[0, 1]$. If $Y = (X_1 + X_2) / 2$, it will also be distributed between 0 and 1, but it won't be uniformly distributed ...
5
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0answers
52 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 ...
0
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1answer
34 views

expectation and geometrical probability problem [closed]

A horizontal line of length $5$ units is divided into two parts. If the first part is of length $X$. Find $E[X]$ where $E[\cdot]$ stands for expectation. how to approach this question ? X can take any ...
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0answers
23 views

Finding conditional probability distribution (X|Y) from (Y|X)

$(Y,X)$ has a joint distribution where the marginal of $X$ is a standard normal and $Y|X \sim U \left[|X|-\frac{1}{2},|X|+\frac{1}{2}\right] $ where $U[a,b] $ means uniform in the interval [a,b]. How ...
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0answers
33 views

Uniform distribution of points on a cone constrained to a continuous line

I was hanging lights on a Christmas tree yesterday, and thought of a problem , which may have an easy solution - but not one that I can think of off the top of my head. It is posed as follows: ...
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0answers
33 views

Uniform Sampling & CDF inverse

I have a probability exam soon, and our prof told us to study the following question: "Describe a procedure for generating independent identically distributed (i.i.d.) samples of a random variable ...
0
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1answer
42 views

Detecting corrupted data in birthdates of a population

I have a population of N birthdates. Let's assume that birthdates are uniformly distributed over the year. I'm concerned that some of these records have been corrupted, for example by someone ...
0
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0answers
43 views

Cramer-Rao Uniform Distribution

If my data, $X_i\sim U(0,\theta)$ is iid. What is the Cramer-Roa lower bound for a variance estimator such as the sample variance? $ S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2$ I am stuck ...
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0answers
36 views

Expected value for number of occurences

Let $W$ be a random word made from letters which are in set $K$ (letters are uniformly distributed in $W$) . Suppose also that $W$ has finite length ($\geq 2$) and size of $K$ is finite ($\geq 2$). ...
2
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2answers
81 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 $iid$ $unif(0,1)$, ...
0
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0answers
28 views

Expectations of order statistics of uniform RVs, via exponential formulation

If $U_1,\ldots, U_n$ are i.i.d. uniform random variables, then I know that the order statistics satisfy $$(U_{(1)},\ldots, U_{(n)}) \overset{d}{=} \left(\frac{X_1}{\sum_{i=1}^{n+1} X_i}, ...
0
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2answers
57 views

Probability of random variable with uniform distribution on an interval

Let a random variable X have a uniform distribution on the interval $[0, 10]$. Find $P(X(X + 10) > 11)$ Since X has a uniform distribution, the pdf of X is $$ ...
0
votes
1answer
20 views

Prove that two random variables are dependent

Given two random variables X and Y where X is uniformly distributed on [-1,1] and Y = X^2, prove that these two random variables are dependent. Of course, it's clear that they are dependent. But, how ...
2
votes
1answer
74 views

$X_1$, $X_2$ i.i.d RVs, $X_1$ is uniformly distributed. Show $E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$

Let $X_1$, $X_2$ be two i.i.d. random variables and $X_1$ is uniformly distributed (discrete) on the set $\{1,2,3\}.$ Show that: $$E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$$ Can someone give me ...
1
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2answers
39 views

Convergence weakly to exponential random variable for this r.v.

Can I get some insight of how to solve this problem? Let $X_1, X_2, X_3, ...$ be i.i.d. copies of uniform random variable on $[0, 1]$. Let $M_n = \text{min}_{1\leq i <j \leq n} |X_i - X_j|$. Show ...
0
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2answers
72 views

Uniform Random Variable: Correlation and Independence

Let X be a uniform random variable defined on the interval $(0,1)$. If $Y = 6X^2−6X+1$, compute the correlation of X and Y . Are X and Y independent? Are X and Y uncorrelated? So my work is. $F(X) ...
0
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3answers
38 views

Find cumulative distribution function of uniform distribution

Random variable X has uniform distribution on $[0,1] \cup [2,3]$. Find cdf of variable X. I mean i do not know how to treat this on such strange interval.
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1answer
58 views

Mutual information for a continuous uniform distribution

I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck. I'm working with a real-valued ...
1
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1answer
76 views

IID Uniform probability problem [closed]

Three students independently attempt to solve a problem. Assume that the times taken by each student to solve the problem are iid according to U(0,30). Find the probability that the student who fi ...
1
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0answers
21 views

Covariance of uniform random and indicator function dependant on it

Define $I = \begin{cases} 1,& \text{if } X\leq a\\ 0,& \text{if } X\gt a \end{cases}$ $X$ is uniform on $[0,1]$. We want to compute $Cov(I,X)$ which involves $E[IX]$. $E[IX] = ...
0
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0answers
22 views

Find probability from Uniform Distribution [duplicate]

Two numbers are selected at random from the interval (0,1). If these numbers are iid and uniformly distributed, find the probability that the three line segments found by breaking interval into three ...
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 ...
0
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2answers
49 views

The distribution of the sum and difference of independent uniformly distributed variables

Suppose $X$ and $Y$ are independent uniformly distributed on the interval $[-a/2,a/2]$. What is the density function of $Z=X+Y$ and of $Z=X-Y$? I know that it will be the convolution of ...
1
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1answer
44 views

Uniform Probobility Distribution Word Problem, grocery store checkout and meeting a friend

The amount of time spent waiting in line at a grocery store express checkout varies from 5 minutes to 15 minutes and follows a uniform distribution. Let X be the amount of time spent waiting in line. ...
4
votes
0answers
86 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 ...
2
votes
0answers
57 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...