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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4 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 ...
2
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1answer
21 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 ...
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2answers
37 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 ...
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0answers
9 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 ...
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0answers
25 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=$X_1$+$X_2$) ...
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0answers
30 views

Generating a Uniform R.V with specified correlation [on hold]

I understand that it involves copulas, but I'm looking for a specific methodology for a specific correlation. I want to generate $U$ and $V$, random variables that are $~Uniform (0,1)$ with ...
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0answers
45 views

Cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$?

I am trying to argue that the cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$. Below the argument I have developed. ...
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1answer
29 views

Product of exponentially distributed and uniformly distributed random variables [on hold]

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 ...
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2answers
25 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 ...
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0answers
94 views

Mathematics Homework 2 Question 8d :What is the probability you and your partner are now able to meet the new deadline? [closed]

You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned ...
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2answers
33 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 ...
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0answers
31 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 ...
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2answers
53 views

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

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

Relation between uniform distribution and dense curves

I have a question on the relation between space-filling curves and the joint distribution of two independent uniform random variables. Consider the probability space $(\Omega, \mathcal{F}, ...
2
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1answer
30 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 $$ ...
1
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2answers
51 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 ...
0
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1answer
21 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 > ...
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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
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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 ...
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0answers
32 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 ...
2
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0answers
38 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 ...
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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: ...
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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 > ...
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2answers
110 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 ...
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0answers
28 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 $𝑓_{𝑌_1,𝑌_2} (𝑦_1, 𝑦_2)$ ...
0
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1answer
31 views

Calculating the mean and variance of continuous distribution

The main question was "A machine produces 2mm to 12mm usb sticks. Any usb greater than 10mm in size will need to be thrown away." Part A) Calculate the portion that needs to be thrown away, and I got ...
2
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1answer
24 views

Properties of the solution of a linear system with random equations

$x_i$ is drawn from $\mathrm{unif}(a,b)$, $y_i$ is drawn from $\mathrm{unif}(c,d)$. $x_i$ are independent from each other. $y_i$ are independent from each other. $x_i$ are independent of $y_i$. $i$ ...
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0answers
26 views

Is the CDF of a mixture distribution uniformly distributed?

It is well-known that if $Y = F(X)$, such that $F$ is a continuous and a strictly increasing cumulative distribution function with a well-defined quantile function $F^{-1}$, then $Y \sim U(0,1)$. Now, ...
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0answers
22 views

Repetitions required to choose from a list

Say I have a list of $n$ objects and I randomly choose one item from the list, but did not remove it. What is the method for calculating the probability I have chosen $x$ different items in the list ...
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0answers
46 views

Distribution of $1/U^2$ where $U$ is uniformly distributed on $(-1, 1)$

Suppose $U\sim \mathrm{Uniform}(-1, 1)$. Let $Y =1/{U^2}$. What is the distribution of Y? Here is what I have: $$ \begin{aligned} Y \in [1,\infty)\\ P(Y <y) = P\Big(\dfrac{1}{U^2} < y ...
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1answer
43 views

Help in probability, Difficult Question:// [closed]

Upon testing 80 resistors manufactured by a certain company, it is found that 15 resistors failed to meet the tolerance design specifications a) Construct a 92% two-sided confidence interval for ...
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2answers
32 views

Generating a random variable from a uniform random variable [closed]

I have no idea how to go about doing this. Any help would be much appreciated.
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0answers
22 views

As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? [duplicate]

both for the situation where elements in the matrix are randomly assigned any number and where elements are assigned a value uniformly distributed between 0 and 1. Thanks so much! (This is not a ...
0
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1answer
36 views

Uniform Distribution Question - Help Needed

Let $X_1, X_2, . . . , X_n$ be a random sample from a uniform distribution on $[0, \theta]$. Suppose results $x_1, x_2, . . . , x_n$ are observed. Since $f(x) = 1/\theta$ for $0 \leq x \leq \theta$, ...
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1answer
40 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatθ = X _{(1)} + X_ {(n)}$ for a U(0, θ) distribution comes from? Where: $X _{(1)}$ = min$_i (X_i)$ $X_ {(n)}$ = max$_i (X_i)$ I know it is not the MLE, ...
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1answer
18 views

Maximum-Likelihood-estimator for number of marbles

Let there be n marbels in a box. Each has a unique number from 1 to n written on it. You pick a marble, write down the number and put it back. This process is repeated N times. Give a ...
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1answer
50 views

approximating a uniformly distributed random variable

Suppose that $U$ is a uniformly distributed (continuous) random variable on $[0,1]$. Let's say that I am interested in finding 3 discrete points $u_1,u_2,u_3$ which approximate $U$ in some sense. My ...
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2answers
31 views

Sum of uniform random variable and non-uniform random variable [closed]

Let $G=\mathbf{Z}/p \mathbf{Z}$ where $p$ is prime, $X\in G$ be a uniform random variable and $Y\in G^{*}$ be any random variable. Is it possible to have $Z=X+Y \in G$ with a uniform distribution? ...
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1answer
22 views

Radio communications system - Uniform density question [closed]

Full question: In a radio communications system, the phase difference $X $ between the transmitter and receiver is modeled as having a uniform density in $[—\pi, +\pi]$. Find $P(X \le 0)$ and $P(X \le ...
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1answer
73 views

Random Variables in a Uniform Probability Space

Suppose that $\Omega = \{1,2,3,4,5,6\}$ is a uniform probability space. Now, let $X(\omega)$ and $Y(\omega)$, for $\omega \in \Omega$, be random variables defined as: $$\begin{array}{|c|c:6c|} ...
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2answers
22 views

Simulation methods and generating random variables

Twenty aircraft are sent to bomb a target that is rectangular in shape. It has dimensions 150m by 50m. Each aircraft makes a bombing run along the horizontal x axis and drops one bomb. The point ...
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1answer
45 views

What is E[X]? What is Var(X)?

The number of accidents X that a person has in a given year is a Poisson random variable with mean Y . However, Y ∼ Uniform ([2, 4]). Calculate: (a) E[X] (b) Var(X) Extra My understanding of the ...
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1answer
77 views

Probability that at least two of three uniform random variables~[0 1.5] add up to >2 [closed]

There's a problem I've been stuck on for a while regarding the sum of two uniformly distributed, independent random variables. The problem goes like this: You find some old batteries in a drawer. ...
0
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1answer
48 views

How could I show that some random variables have same distribution?

Let X1, . . . , Xn, be random variables uniformly distributed over the interval [a, b]. Let Y1 < Y2 < · · · < Yn be the same values in sorted order. Let Y0 = a and Yn+1 = b. Show that ...
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0answers
13 views

Analytical expression for expected the n-th and (n-1)th order statistic of minimum of two uniform random variables

I have two independent random variables $X_1 \sim Uniform(b,1)$ and $X_2 \sim Uniform(0,1)$, where $0<=b<1$. Their CDF's are: $$ F_{X_1}(z) = \begin{cases} 0 & z<b \\ \frac{z-b}{1-b} ...
0
votes
1answer
65 views

Assuming the two people wait for each other, what is the expected waiting time?

Two people agree to meet at a restaurant. Assume their arrival times are independent and uniformly distributed on the one hour interval from 1:00–2:00 p.m. Assuming the two people wait for each other, ...
0
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1answer
23 views

Box-Muller method to Polar Marsaglia scheme

I have just learned the Box-Muller method for generating normal random values. My notes then consider the Polar Marsaglia method, which is more efficient than Box-Muller. In Box-Muller: ...
0
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1answer
20 views

How do we know the pdf of $X_{max}$ for a uniform distribution? [closed]

How do we know the pdf of $X_{max}$? We have a sample of size $n=6$ from a uniform distribution of the interval $[0,\theta]$. Let $\hat{\theta}=X_{max}$ be the estimator of $\theta$. (a) Given ...
0
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0answers
33 views

Tukey's symmetrical lambda distribution

U ~ Uniform(0,1) $$Z_\lambda = \frac{U^\lambda-{(1-U)}^\lambda}{\lambda} $$ I have to find the first four moments and two ($\lambda_1,\lambda_2$) such that they have the same four moments.