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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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
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1answer
33 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 $$ ...
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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 $f(x)$...
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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 $Z = \min\{X_i : X_i > c\}$...
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0answers
17 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 $\phi \in [0,2\pi]$. ...
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2answers
52 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'...
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0answers
33 views

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

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

Density probability

I ask myself a question about of density next : p(xi)=1/(pi*(xΒ²+1)) The law marginal is easy to identify of X and Y: ...
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0answers
31 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
112 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
30 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 $π‘Œ_1 = 𝑋_1 + 𝑋_2$, and $π‘Œ_2 = 𝑋_2 βˆ’ 𝑋_1$. a) Find the joint pdf $𝑓_{π‘Œ_1,π‘Œ_2} (𝑦_1, 𝑦_2)$ ...
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1answer
36 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
27 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
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 ...
2
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0answers
47 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 \Big)\\ =...
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1answer
46 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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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 ...
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1answer
37 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
41 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatΞΈ = X _{(1)} + X_ {(n)}$ for a U(0, ΞΈ) 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 maximum-...
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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
34 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 $[β€”\pi, +\pi]$. Find $P(X \le 0)$ and $P(X \le ...
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1answer
75 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
25 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
47 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
78 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
49 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 < Β· Β· Β· < Yn be the same values in sorted order. Let Y0 = a and Yn+1 = b. Show that ...
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0answers
14 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
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1answer
66 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
27 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: $$X=\sqrt{-2\...
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1answer
21 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 $\...
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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.
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0answers
12 views

The expectation number of collision to slow down below certain value

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward ...
0
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1answer
31 views

What is the conditional distribution function of P(T < z| U=x) where U, V are uniformly distributed on [0,1] and T= Max(U,V)

Let U, V be independent random variables, both uniformly distributed on [0,1].T= max (U, V). 1) What is the joint distribution function of T and U, P(T <= Z, U <= x)? I did the following: P(T ...
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1answer
35 views

Understanding Bayes' theorem with uniform distribution

Station $X$ begin to transmit a message in $[0,20]$ with uniform distribution, and $Y$ also want to transmit a message in $[6,14]$ with uniform distribution. Assume that transmission takes $2$ ...
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1answer
23 views

Integrating PDF of Continuous Uniform RV to get CDF

For a continous uniform RV $$\text{PDF} = \frac{1}{b-a} \text{ for } x\in[a,b)$$ (sorry cant figure out how to add whitespace) and $$\text{CDF} = \int_{-\infty}^x \text{PDF} \, dx$$ so for $x>0$ ...
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26 views

Distribution of a function of Discrete Uniform RVs

Suppose $X$ and $Y$ take the values ${0,1,...,N}$ with probability $\frac{1}{N+1}$. What is the PMF of $|X-Y|$? I started it like so: Let $|X-Y| = Z$ $P(Z<z) = P(-z < X-Y<z)$ In the ...
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0answers
16 views

Targetting the distribution of distances between points

For a certain problem, I need to make distance dependent statistics, but with the constraint that the number of sampling points, $N$, should be kept as small as possible. To be more specific I need to ...
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1answer
53 views

Find the joint pdf of X and Y for two uniform distributions

Let X have a uniform distribution on the interval $(0,1)$. Given that X = x, let Y have a uniform distribution on the interval $(0,x+1)$. Find the joint pdf of X and Y. Sketch the region where $f(x,...
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0answers
20 views

evaluation of this integral arising in probability theory

Let $U_1,...,U_n$ be independent, uniformly distributed RVs on $[0,n]$. Consider the ordered statistics of these, $$U_{1:n} < ... < U_{n:n}$$ So $U_{1:n} = \min\{ U_1,...,U_n\}$ and so on. ...
3
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1answer
40 views

Average waiting times

I have the following exercise, which I would like to solve: Company A run buses between New York and Newark, their bus leaves New York every half an hour starting from 0:00, 24h a day. Company B also ...
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22 views

Uniformly Random Polynomial

Hypothesis: all the polynomials and values are defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number. My question is related to information security. Assume (in a protocol) ...
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9 views

Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: \begin{equation} b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m. \end{equation} We can include in those ...
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29 views

Convolution of a Pareto and a Uniform distribuion

I would like to convolve a Pareto and a Uniform Distribution. Pareto's PDF: $f_X(x)=\begin{cases} 0 & x < c \\ b\frac{c^b}{x^{b+1}} & x\geq c \end{cases}$ The $[0,a]$ ...
3
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1answer
37 views

Expected value of a series of random variables

There are $K$ checkout counters in the mall, and there are $N$ shoppers in the queue waiting for a checkout counter. Initially all counters are empty. Whenever a counter is empty, the next shopper in ...
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0answers
25 views

Working out probability distribution from given uniform random variable

I'm not able to understand how they got from probability U is less than e to the power of x to e^-x? shouldn't it be e^x instead because we are given in our notes that probability U that's less than u ...
0
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1answer
21 views

does calculating the arithmetic mean of a serie of standar deviation measures make sense?

I'm trying to test if a hash function mantains a even distribution of values. My plan is to generate a set X of hashes and a integer Y symbolizing "slots" to see how many of the hashes maps to the ...
8
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0answers
79 views

Picking points on a sphere at random

Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that ...