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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2answers
49 views

Why birthday distribution is not uniform.

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
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0answers
18 views

Probability density function of an uniformly distributed random variable [on hold]

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1).$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ for $\lambda>0$ And calculate $P(Y>t+s|Y>t)$ for ...
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3answers
27 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
3
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1answer
36 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
2
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1answer
32 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
2
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0answers
25 views

Is there a connection between uniform law of large number and Ibragimov's conjecture?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following: Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, ...
2
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0answers
33 views

Non-Linear System of uniform distributions. Determine the Density functions.

Consider the non-linear system: $$ Z = -X + W\\ Y = X + XV. $$ Where $X$, $V$ and $W$ are mutually independent and all are $\sim U(0,1)$. I have got some problems finding the distributions of the ...
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2answers
34 views

Uniform distribution on $\{1,\dots,7\}$ from rolling a die [duplicate]

This was a job interview question someone asked about on Reddit. (Link) How can you generate a uniform distribution on $\{1,\dots,7\}$ using a fair die? Presumably, you are to do this by combining ...
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1answer
27 views

Uniform distribution on sphere

Let $U = (u_1, u_2, u_3)$ is random vector uniformly distributed on unit sphere $S^{2} \subset \mathbb{R}^3$. Are $u_1, u_2, u_3$ mutually independent ? I guess not, but I have no idea to prove it.
0
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1answer
31 views

Distribution of the difference of order statistics

Let $X_1$ and $X_2$ be a random sample of size 2 from a uniform distribution over the interval $[0,1]$, let $Y_1$ and $Y_2$ be the corresponding order statistics. Find the conditional density ...
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2answers
56 views

The probability density of $X^2$?

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? ...
0
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1answer
25 views

Calculate $P(|X-4| > 1.5)$

If $X \sim U(2, 8)$ Would it be $$P(X > 1.5 + 4) + P(X <-1.5 +4)$$
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0answers
34 views

What is the Fourier transform of a uniform random process U(0,1)

What is the Fourier transform of a discrete uniform random process $U(0,1)$? The discrete uniform random process could be for example $U(0,1)=0,0,1,1,1,1,0,0,1,0,0,0,0,0$ of any length $N$. Is there ...
0
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0answers
20 views

Exlain the significance of the uniform random variable for the simulation of random variables

I can think of the "Universality of the Uniform": Given an Unif(0,1) r.v., we can construct an r.v. with any cts distribution we want. Conversely, given an r.v. with an arbitrary cts ...
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1answer
19 views

unbiased estimator of $m^2$ of uniform distribution over $(0,m)$ [closed]

I have a sample of size $1$ from a distribution that has $U\left(4,4+m\right)$ as probability density function. Is there any unbiased estimator for $m^{2}$?
2
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0answers
29 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
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1answer
40 views

Limit moment generating function

For n a natural number let $X_{n}$ have discrete uniform distribution on interval {1,2...,n} and $Y_{n} =\frac{1}{n} X_{n}$. I need to show that for all t(real number) the $\lim_{n \to \infty} ...
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1answer
53 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
0
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1answer
61 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
0
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0answers
29 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
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2answers
54 views

Average distance from center of circle to evenly-distributed points within it

With some number of points that are evenly/uniformly (assuming those mean the same thing) distributed within a circle of radius 1, what is the average distance from the center of the circle to a ...
0
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0answers
45 views

Expectation of $v=\inf \{n\geq 2\,;\, X_n > X_1 \}$ when $(X_n)$ is i.i.d. uniform on (0,1)

Let $W$ be the occurence meaning the following ordering : $X_1...X_k$ where $X_k$ is greatest.. $X_k$ is greatest, and next in order is $X_1$, and the order of the others is not important. Because of ...
2
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2answers
41 views

Maximum likelihood estimator on uniform distribution

I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me mad. Here it goes. Consider the uniform distribution on ...
2
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1answer
23 views

Maximum likelihood on uniform distribution

In a exercise i'm doing it is asked to find the maximum likelihood estimator of a random sample $X_{1}, ... , X_{n}$ of a population with distribution $X\sim U(- \theta , \theta) $. I've found that ...
2
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2answers
45 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
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1answer
37 views

Bivariate probability distribution(s) over unit square, uniform marginals, midpoint is saddlepoint

Construct a bivariate probability distribution--or family of such distributions--over the unit square (corners $(0,0), (0,1), (1,1), (1,0)$) with uniform marginals and having a saddlepoint at ...
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1answer
42 views

Distribution of function of a Random Variable

If $X$ is uniform on $(0,1)$, how would I go about finding the CDF of $Y=(X-X^2)^2$ ?Thanks.
1
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1answer
82 views

Uncertain about which probability method to use for the problem

Suppose I want to catch a bus (which runs every 10 minutes on average). What is the probability that: 1). You will wait for at least fifteen minutes before the bus arrives, and then, 2). 3 buses ...
0
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0answers
26 views

Conditional expectations with identical marginals and positively dependent but unknown joint distribution

Let $A$ and $B$ be random variables, each with marginal distribution $% U\left( 0,1\right) $, but unknown joint distribution $H\left( a,b\right) $. Suppose $A$ and $B$ are each stochastically (weakly) ...
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0answers
5 views

Sampling specific unit vectors

Given a unit vector $A\in \Bbb{R}^N$ and an angle $\theta$, the unit vector $P$ needs to satisfy $\left<A,P\right>=\cos\theta$. How to sample $P$ uniformly? For example: If $A = ...
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1answer
21 views

Optimal Value & Uniform Distribution

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$. I want to find optimal d in this expression, $aR-(d^2-1)/2$. When I try to find out optimal $d$ than it is $0$. ...
2
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0answers
34 views

Inconsistent answers with conditional expectations

Let $X$ and $Y$ be two independent random variables distributed uniformly on $[0,1]$. I want to compute: $$E[X+Y\mid\max\{X,Y\}≤(1/2)]$$ My first approach was the following. Let $X=\max\{X,Y\}$. ...
2
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2answers
77 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
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2answers
52 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
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0answers
13 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
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1answer
38 views

Demonstrate uniform continuous distribution using tangible items?

What is the best way to explain "equally likely" in continuous uniform distribution to an audience using tangible or everyday items?
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2answers
66 views

Calculating if a point is within the overlap of two circles

Two circles of equal radius (R) intersect as shown below. Assuming more points are uniformly distributed in an area with dimensions D*D, where D = 4*R. What is the probability that a point will be ...
2
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1answer
32 views

Stoppage time for sequence of uniform random numbers with a recursively shrinking domain

Define $x_n = U(x_{n-1})$ where $U(x)\in\lbrace 0,1,\ldots,x\rbrace$ is a uniformly distributed random integer. Given $x_0$ as some large positive integer, what is the expected value of $n$ for which ...
0
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1answer
26 views

How were the bounds of integration changed here in this probability problem on uniform distributions?

I have the following problem and solution: I don't understand how the bounds of integration were changed from 0 to 1, to $x^2$ to 1. I see where $1/\sqrt{y}$ was substituted in for $f(x|y)$ and 1 ...
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1answer
25 views

calculating probabilities about the sum of dependent discrete uniforms

Say I have the following information: $$ X_i \sim \text{Discrete Uniform}(1,13) $$ and I want to find $\mathbb P(X_1+X_2+X_3 \ge 25)$ for the cases where the $X_i$'s are dependent. What approximations ...
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2answers
46 views

Probability Question About Uniform Random Variables and Median

Let U, V, W ∼ Uniform(0, 1) be independent. Find the probability that the median (i.e., the second smallest) of these three random variables lies in the interval (1/4, 3/4). I cannot figure out what ...
0
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1answer
163 views

uniformly distributed random variables

Ram and Shyam wanted to meet at a park about 12.30 P.M.. If Ram arrives at a time uniformly distributed between 12.15 P.M. to 12.45 P.M. and if Shyam independently arrives at a time uniformly ...
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2answers
93 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 ...
2
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2answers
55 views

Mean of the difference between uniform random variables.

I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then ...
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1answer
46 views

interval learning unbiased estimator

Suppose that I draw $n$ points uniformly at random from $\mathcal{U}(-a,a)$ for some $a\in\mathbb{R}_+$. Denote this set of points $\{X_i\}_{i=1}^n$. Now for some $-a < x_1 < x_2 < a$, let a ...
2
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1answer
57 views

Indicator function and expectation

Consider a random variable $X$ uniformly distributed over $[0,1]$ and the indicator function $\mathbb{1}_{X \geq \tilde{x}} $ equal to one if $X \geq \tilde{x}$ and zero otherwise. We know that ...
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1answer
47 views

Conditional Probability Distribution for two Discrete Uniform Random Variables with given Correlation Coefficient

I consider a problem with two random variables $X, Y \sim Unif\{a,b\}$, for which I want to set a correlation coefficient $Corr(X,Y)=\rho$. Now, I am interested in the conditional probability mass ...
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2answers
40 views

How to prove that equation over probabilities has unique solultion or find counterexample?

Given equations: $$ \prod_{i=1}^n p_i = \prod_{i=1}^n (1-p_i)= \frac{1}{2^n} $$ where $p_i\in (0,1), i=\overline{1,n}$. Is it true that this system has unique solution $p_1=p_2=\ldots=p_n=\frac12$ ...
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4answers
85 views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that ...
2
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1answer
38 views

Calculating power of a Hypothesis Testing Problem based on Uniform distribution

Consider the problem of testing $H_0:a=0$ against $H_1:a=1/2$ based on a single observation X from U(a,a+1). The power of the test "Reject $H_0$ if $X>2/3$" is (A)1/6 (B)5/6 (C)1/3 (D)2/3 ...