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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3
votes
2answers
486 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 ...
0
votes
1answer
40 views

Show that $\mathbb{P}(|\frac{1}{n}\sum_{i=1}^nX_i-\frac{1}{2}|>\frac{1}{2})\leq\frac{1}{3n}$

Suppose $X_1, X_2, ..., X_n$ are independent uniformly distributed random variables on [0,1]. Show that $\mathbb{P}(|\frac{1}{n}\sum_{i=1}^nX_i-\frac{1}{2}|>\frac{1}{2})\leq\frac{1}{3n}$ I've ...
0
votes
1answer
33 views

How do I evenly distribute some time? [on hold]

Here's my problem. I have a device (ozone generator) which produces 10g of ozone per hour, or 2.77778 milligrams per second (I think). I need to be able to control the production per hour by pulsing ...
-1
votes
2answers
42 views

Probability of even sum of $n$ integers with uniform distribution from $\{1,2,\dots, 2n\}$.

Choosing with Uniform distribution $n$ numbers from $\{1,\dots,2n\}$ with returns and the order is important. What is the probability that the sum of these number will be even? Thanks.
0
votes
1answer
34 views

Probablity of normal distribution when x is a function

Assume a uniform distribution random variable X~U(0,1). And $\Phi$ is the symbol of the standard normal distribution. Assume $Y=\Phi^{-1}(X)$. The question is, $\mathbb{P}(Y \le 0)=?$. The Solution is ...
1
vote
1answer
38 views

Let $E$ := {$U_1 \geq U,U_2 \geq U,U_3 < U,U_4 \geq U, U_5 < U,U_6 \geq U,U_7 \geq U$}

Let $U,U_1,U_2,...$ be independant, on [0,1] uniform distributed random variables. Let $E$ := {$U_1 \geq U,U_2 \geq U,U_3 < U,U_4 \geq U, U_5 < U,U_6 \geq U,U_7 \geq U$}. Find the probabiliy $...
0
votes
1answer
23 views

Method to find out how distributed are a certain set of data?

Assume I have array of $A_{3\times120}$ Each row of matrix A corresponds to a shape which is generated by its three row elements as below: $r=1+a_1\cos(\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)$ ...
1
vote
1answer
27 views

Joint density function of $T_1,T_2$ and expectation of $E[T_1 ^2 +T_2 ^2 ]$

Given that $T_1,T_2$ are random variables representing the useful life (in hours) of two electrical appliance. The joint probability function of two variables distributed uniformly in the domain ...
1
vote
3answers
39 views

Distribution of ages of 3 children in a family

Please consider the following problem: A family has 3 children, creatively named A,B, and C. (a) Discuss intuitively (but clearly) whether the event “A is older than B” is independent of the event “...
6
votes
1answer
84 views

Order statistics for discrete uniform random variables

Let $X_i, i=1,\cdots,N$ be i.i.d. discrete uniform random variables, taking values in the range $\{0,1,...,M-1\}$. Let $X_{(i)}$ denote the $i$-th order statistic. What are the values of $\...
1
vote
2answers
26 views

Uniform Distribution Problem on $X, Y, Z$

Problem: Let $X \sim \text{Uniform}(0,1)$. Let $0 < a < b < 1$. Let $$ Y = \begin{cases} 1 & 0 < X < b \\ 0 & \text{otherwise} \end{cases} $$ ...
0
votes
1answer
26 views

Probability that a set of uniformly distributed random variables is 'greater' than another such set.

Suppose we generate several uniformly distributed random variables (between 0 and 1), and arrange them in descending order to form a set [A1, B1, C1...]. We then do the same process to form a second ...
1
vote
1answer
46 views

Determine $\lim_{n\to\infty} \mathbb{P}(\sum_{i=1}^n X_i \leq \frac{n}{2})$

Suppose $X_1, X_2, ... X_n$ are independent and uniformly distributed (on $[0,1]$) random variables. Determine $\lim_{n\to\infty} \mathbb{P}(\sum_{i=1}^n X_i \leq \frac{n}{2})$ My thoughts were the ...
0
votes
1answer
28 views

Distribution function of uniform distribution

I don't know why the distribution of this question is that when x is in between 0 and theta. In solution.. is that right? I searched the distribution of uniform distribution. But it is alike with that ...
0
votes
0answers
30 views

Joint pdf of two uniform random variables on a unit line segment

Let $X$ be a standard uniform random variable, define $Y=1-X$. Then supposedly $X$ and $Y$ are uniform over a 1-simplex, so their joint distribution should be Dirichlet of order $K=2$, and $\alpha_1=\...
0
votes
0answers
27 views

How to define a uniform probability distribution over a convex polytope / polyhedra and add them?

Let $P$ be a convex 3d polyhedra / 2d polytope constrained by a set of linear inequalities $Ax<= b$. 1.How to define a uniform probability distribution over a polytope/polyhedra? Let us say we ...
5
votes
0answers
32 views

$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...
1
vote
2answers
17 views

Integration Question Arising from Attempting to Compute a Marginal Distribution

I have a question about the last step of a proof on page 37 of All of Statistics. The entire proof is here for sake of completeness, but I don't think grokking it in its entirety is necessary for ...
0
votes
1answer
1k views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, $...
1
vote
1answer
21 views

Mathematical formula for equal distribution of amount among different group [closed]

Please let me know if you think i should edit my question or description. Problem statement: lets say i have spent $x on a sports material which needs to shared among total y memeber of the team. But ...
2
votes
2answers
26 views

Find function $h$ so that $h(U,V)$ equals density of $f(a) da$ for $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$ and let $U,V$ be independant and uniform distributed on [0,1]. Now I want to find a function $h$ so that $h(U,V)$ is equal to the density $f(a)...
1
vote
1answer
60 views

is arcsin() evenly distributed?

Given a uniform random distribution P of real numbers from [0,1] how might I prove (or disprove) that the map from P to Q of $(p\in{P} \rightarrow q=arcsin(p)\in{Q}) $ is a uniform distribution over [...
1
vote
1answer
18 views

Probability of not choosing from a set of replaced values

The problem statement is as follows: There is a set of numbers N numbers $1..N$ (eg: N = $10^6$) N numbers are chosen uniformly and independently with replacement I would like to be able to ...
2
votes
1answer
807 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
2
votes
2answers
24 views

Getting the marginal distribution from the joint pdf

To bein with, I did the following calculations: $$ Y\sim Uniform(0,x)\\ f_x(x)=\{\frac{1}{x^2},x\ge1\}\\ f_{y|x}(y)=\{\frac{1}{x},0\le y \le x\}\\ f(x,y)=f_x(x)f_{y|x}(y)=\frac{1}{x^3},x\ge 1,0\le y\...
1
vote
1answer
37 views

Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...
4
votes
2answers
57 views

Meeting probability of two bankers: uniform distribution puzzle

Two bankers each arrive at the station at some random time between 5PM and 6PM (arrival time for each of them is uniformly distributed). They stay exactly five minutes and then leave. What is the ...
0
votes
0answers
33 views

Find continuous stochastic variable $X$ with PDF $f_X = \frac{1}{x^2}$

Given the uniform stochastic variable $U$ defined on the interval [0,1]. Using $U$, define a continuous stochastic variable $X$ with probability density function (PDF) $$f_X(x) = \begin{cases} \frac{1}...
2
votes
1answer
29 views

Computing the distribution of a uniform r.v. with parameter being another uniform r.v.

I have this: Let $X\sim U(0,1)$, $Y\sim U(X,1)$. What is the distribution of variable $Y$? My answer: I use a geometric approach since everything happens in the square $(0,1)\times (0,1)$, see ...
1
vote
2answers
44 views

A stick is broken into two pieces, at a uniformly random chosen break point. Find the CDF.

I'm having trouble understanding how the CDF is found in the solution below: We can assume the units are chosen so that the stick has length $1$. Let $L$ be the length of the longer piece, and let ...
0
votes
0answers
33 views

What distribution results from drawing random numbers whose upper bound is normally distributed?

I have a normal distribution $N$ with $μ=U/2$ and $σ=U/12$ (an approximation of the Irwin-Hall distribution) which has been bounded and normalized to $[0,U]$. I will now repeatedly generate random ...
1
vote
0answers
10 views

Formula to evenly distribute elements without knowledge of the other buckets

I am trying to write a formula to determine how to evenly distribute elements into individual buckets without specific knowledge of each bucket. The only knowledge that you have is the max number of ...
1
vote
0answers
26 views

PMF for sum of uniformly distributed random variables

Let $X_1$ and $X_2$ be independent integer valued random variables that both are uniformly distributed on {1, 2, . . . n}. What is the PMF for S := $X_1$ + $X_2$? What I have so far: P(S=$X_1$+$X_2$) ...
1
vote
2answers
53 views

If $f(x)$ is a strictly increasing function on the unit interval, what is the distribution of $f(\mathcal{U})$? Prove it.

$\mathcal{U}$ is distributed uniformly on the interval $[0,1]$. If $f(x)$ is a strictly increasing function on the unit interval, what is the distribution of $f(\mathcal{U})$? Prove it. Well if $f(x)$...
0
votes
2answers
29 views

Uncorrelated but not independent uniform distribution

Let $X = (X_1, X_2)$ be uniform distributed on $\{(-1,0), (1,0), (0,-1), (0,1)\}$. First of all I want to show that $X_1$ and $X_2$ are uncorrelated but not independent. Secondly I thought about ...
0
votes
1answer
30 views

Product of exponentially distributed and uniformly distributed random variables [closed]

Let $X$ be an exponentially distributed random variable, and let $V$ be a uniformly distributed random variable on $\{-1,+1\}$ that is independent from $X$. Furthermore, let $Y = X \cdot V$. I want ...
1
vote
2answers
55 views

How do I find the cdf of $X_1 + X_2$?

$X_1$ uniform $(0,1)$ and $X_2$ uniform $(0,2)$ $$ \begin{cases} f(x_1,x_2) = \frac{1}{2}, &\quad \mbox{for} \ 0<x_1<1, 0<x_2<2 \\ 0, & \quad \mbox{otherwise} \end{cases} $$ ...
1
vote
3answers
169 views

Expected value of maximum of three random variables from uniform distribution

Three uniform random variables $X = [2.9,3.1]$, $Y = [2.7,3.1]$, $Z = [2.9,3.3]$. What is the expected value of the maximum of these three variables? $E(\max(X,Y,Z))$. I have tried to split the ...
0
votes
2answers
36 views

Computing the probability of waiting someone - Uniform distribution

I have the following problem and I having trouble in finding it solution. I need a hint. The problem: Two people arranged to meet between 12:00 and 13:00. The arriving time of each one is i.i.d. and ...
1
vote
0answers
36 views

Statistics $X_{(1)}$ complete for a Uniform Distribution?

Someone had asked this earlier, but since it was good practice for my qualifying exam coming up, I figured I would ask and share my work on the problem. The problem is: Suppose $X$ is Unif$(0,\...
0
votes
0answers
17 views

why the uniform distirbution function F(X) equal to 1 when the X is a fixed value?

I have the following quetion: Let X be a continuous random variable with distribution function $F_X(x)$ and density function $ f_X(x)$. Consider the random variable Y dened by $Y = X $ if $X < a$ ...
2
votes
1answer
31 views

Why the probability distribution of a uniform random variable is the Lebesgue measure?

Consider the random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ distributed as a uniform on $[0,1]$. The probability distribution function of $X$ is defined as a map $$ ...
0
votes
1answer
22 views

Uniform random variables, length of smallest interval around given point

The following claim is stated without proof/reference in something I am reading: Let $X_1,\ldots,X_n$ be i.i.d. uniform on $[0,1]$, and let $c \in (0,1)$ be fixed. If $Z = \min\{X_i : X_i > c\}$...
2
votes
2answers
111 views

Choose $x$ objects without replacement from a bag with $n$ object.

General problem: Suppose there is a bag containing $n$ items with $m$ unique values $(m \leq n)$. The distribution of values across all the items is uniform. How many unique values I most probably ...
0
votes
0answers
15 views

Points in hemisphere over plane defined by a normal vector

I have the following formulas to sample points uniformly on a unit sphere in 3D space: $x = \sqrt{1-u^2} sin\phi$ $y = \sqrt{1-u^2} cos\phi$ $z = u$ where $u \in [-1,1]$ and $\phi \in [0,2\pi]$. ...
2
votes
2answers
51 views

Probability of rectangles area being less than 0.5 w/ total length of sides = 2

Question: A random point splits the interval [0,2] in two parts. Those two parts make up a rectagle. Calculate the probability of that rectangle having an area less than 0.5. So, this is as far as I'...
2
votes
0answers
40 views

Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\rightarrow 0$ in probability.

This is a qual problem。 Let $n$ points be iid uniformly distributed on the unit circle. Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\...
0
votes
0answers
25 views

Density probability

I ask myself a question about of density next : p(xi)=1/(pi*(x²+1)) The law marginal is easy to identify of X and Y: ...
0
votes
0answers
29 views

Find Expected Value, Variance, and Limit of Uniform Distribution

Let $X_1, X_2, \ldots, X_n$ be a sequences of independent random variables. $X_i \sim U(0, 2A)$. Compute $E(X_i)$ and the $Var(X_i)$. Also compute the $lim_{n\to\infty} P(X_1, X_2, \ldots , X_n > ...
0
votes
0answers
29 views

joint pdf for two independent uniform distribution

Suppose that $𝑋_1$ and $𝑋_2$ are independent and follow a uniform distribution over $[0, 1]$. Let $𝑌_1 = 𝑋_1 + 𝑋_2$, and $𝑌_2 = 𝑋_2 − 𝑋_1$. a) Find the joint pdf $𝑓_{𝑌_1,𝑌_2} (𝑦_1, 𝑦_2)$ ...