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Questions tagged [uniform-distribution]

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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1answer
29 views

Probability of nth event is between x and y

I have a uniform distribution of age in the range $[a, b)$ with $a=42$ and $b=78$ So the probability that a person walks in a bank that is between $50$ and $70$ years of age would be $\frac{70-50}{78-...
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1answer
210 views

Hypothesis Testing under Uniform Distribution Question

The question reads: Let $\theta > 0$ and $X \sim \mathcal{U}[0, \theta]$, i.e. $X$ is uniformly distributed on the interval $[0, \theta]$. Assume that $\theta$ is unknown, but we can observe $X$. ...
4
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2answers
121 views

Uniform distribution variable in the Newton symbol

The following question is from actuarial exam. Let $N$ be uniformly distributed on $\{0,1,2,...,19\}$. Compute $$\mathbb{E}\sum_{k=0}^{N}{N-k \choose k}(-1)^k$$ I started $$\mathbb{E}\sum_{k=0}^{N}{N-...
2
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2answers
858 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 $[0,...
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0answers
11 views

Expected number of neighbors in an Area with Random Uniform distribution

In an area of ${100m^2}$ 19 sensor nodes are uniformly and randomly distributed. Root node is located at the center of this area. Each node has a transmission range of 40 meters in all directions. All ...
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0answers
20 views

MLE for special uniform case [duplicate]

I'm preparing for my exam and I came across this question: Let $X_1, X_2, ...,X_n$ be iid with PDF $f(x)=\frac{2x}{\theta^2}$ for $0 \leq x \leq \theta.$ Find the MLE of $\theta$ So this is what ...
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1answer
45 views

If $X_1$ and $X_2$ are uniformly distributed random variables with parameters $0$ and $1$, what is the distribution of $Y = X_1 + X_2$?

I was doing a recap of the probability theory I had last year and even though this question shouldn't be hard, it is somehow confusing me immensly. Clearly, if we have $X_1, X_2$ belonging to $U[0, ...
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1answer
33 views

Method of Moments estimators of $\alpha$ and $\beta$

Let 5 numbers 2, 3, 5, 9 and 10 come from a uniform distribution on the interval $[\alpha,\beta]$. Find the method of moments estimators of $\alpha$ and $\beta$. Any help would be appreciated, thank ...
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1answer
868 views

Uniform distribution

You arrive at a bus stop at 10'0 clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. what is the probability that you wait longer than 10 minutes? if at 10:...
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1answer
110 views

MLE of $\theta$ in $U[0,\theta]$ distribution where the parameter $\theta$ is discrete

Consider i.i.d random variables $X_1,X_2,\ldots,X_n$ having the $U[0,\theta]$ distribution: $$f_{\theta}(x)=\frac{\mathbf1_{[0,\theta]}(x)}{\theta}$$ , where the unknown parameter $\theta\in\{1,2,\...
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1answer
18 views

Would this method approximate a uniformly distributed random points on sphere?

I know there are several ways to generate uniformly distributed random points on the 2-sphere $S^2$. But I would like to know if my method does the same job, although it is very inefficient. Say, I ...
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28 views

What is the name of this principle?

When generating uniformly-distributed samples from a multidimensional distribution, I believe that sampling each dimension independently produces uniformly-distributed samples from the original ...
5
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1answer
2k views

Proof that normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = \left(\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \...
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1answer
82 views

Use the change of variables to determine the density for a uniform distribution on $[a,b]$

Knowing that the density of a uniform random variable on $[0,1]$ is: $f_{U}=\left\{\begin{matrix} 1 & x\in [0,1]\\ 0 & x\notin[0,1] \end{matrix}\right.$ How to determine the density of a ...
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2answers
96 views

Pdf of $X+Y+Z$ where $X,Y,Z$ are independent $U(0,1)$

This is my working out of the problem so far, I want to know if there is a more simpler way to solve this, or I would just be interested in other methods that one could use to solve a similar problem
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0answers
22 views

PDF of conditional uniformly distributed random variable

Given two barrels of water, $A,B$, with 1 liter each. We pour an $X\sim U[0,1]$ amount of water from $A$ to $B$ and then $Y$ amount of water randomly from $B$ to $A$ $(Y|X=x\sim U[0,1+x])$. ...
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0answers
18 views

Prove convergence of a sum of random variables

I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem ...
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0answers
9 views

Average Number of users based on some condition and error calculation

I want to Calculate average No. of users selecting any particular No from random number range (2 to 6) and I have defined error condition as when more than one user selects same number. Now based on ...
2
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3answers
76 views

Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases?

Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases? I first found the density functions: $$ f_x(t) =\begin{cases} 1 &...
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1answer
30 views

The probability of sum $x+y$ to be greater than $20$

The variable $x$ takes a value between $0$ and $10$ with uniform probability distribution.The variable $y$ takes a value between $0$ and $20$ with uniform probability distribution. The probability of ...
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2answers
29 views

Distributing 6 persons in their shifts evenly with rotating partner

I thought this was easy but I was mistaken. I need to make a schedule for 6 persons. They have to make shift by two (partner) but the partner should also rotate so that all person gets to be ...
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1answer
27 views

Self-study Order Statistics

So I got this exercise from a book and I'm confused by a statement they made. Example: In a 100-meter Olympic race, the running times can be considered to be $U$~$(9.6, 10.0)$-distributed. Suppose ...
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1answer
272 views

Is modular multiplication under a prime modulus uniformly distributed?

Given a prime $p$ and $m \in Z_p^*$. Assume we draw $a \stackrel{u}{\in} Z_p^*$ uniformly at random. Will $a \cdot m \; mod \; p$ be distributed uniformly over $Z_p^*$?
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136 views

Doubt in proof for Complete Statistic for Uniform Distribution

A statistic $T(X)$ is called complete statistic for a parameter $\theta$, if $E_{\theta}g(T) = 0$ for all $\theta$ implies $P_{\theta}(g(T) = 0) = 1$ for all $\theta$. I interpret $P_{\theta}(g(T)...
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2answers
48 views

Uniform Distribution - Is my solution correct?

I'm preparing for an exam in Probability and Statistics and since I am self-studying, I don't have answers to refer to. So would be really great if someone would please take the time and check if my ...
2
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1answer
61 views

$U_1,U_2,…$ i.i.d. $U[0,1]$, $P\sim \mathrm{Poi}(\lambda)$, find $F_{\operatorname{min}(U_1,…,U_P)}$

Let $(U_n)_n$ a sequence of random variables i.i.d $U[0,1]$ and let $P\sim \mathrm{Poi}(\lambda)$ a random variable such that $P$ is independent of $(U_n)_n$. Let $$ \\ X=\left\{\begin{matrix} \...
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1answer
40 views

Uniform Distribution: Expectation $E(\bar{X})$

Uniform Distribution: Expectation $E(\bar{X})$ Assuming that the samples are independently and identically distributed (iid): First i was asked to find $E(X_k)$ $E(X_k)= $$\int_{0}^{θ} dn P_θ(n) n =...
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1answer
35 views

Generating a number belonging to N(0,1) using *m* numbers from U(0,1) using central limit theorem

I was going through a blog which details how to generate a multivariate Gaussian vector, given a mean vector μ and co-variance matrix σ. As a starting point, author uses generated uniform ...
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0answers
91 views

Show that $P(N \geq n) = (1-e^{-\lambda})^n/ \lambda^n $

The lifetime $X$ (in days) of a device has an exponential distribution with parameter $\lambda.$ Moreover, the fraction of time which the device is used each day has a uniform distribution over the ...
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2answers
72 views

find the probability about sum of random variables

Let $X_1, X_2, X_3, Y_1, Y_2, Y_3, Z_1, Z_2, Z_3$ be random variables which have uniform distribution between 0 and 1. It means, the average of $X_1 = 0.5$ Let: $X=X_1 + X_2 + X_3,$ $Y=Y_1 + Y_2 + ...
55
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1answer
1k views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{...
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1answer
26 views

Convergence of sequence of uniforms

Let $X\sim\mathrm{ Uniform}(0,1)$. Consider the sequence $X_n = X^n$. I want to study the convergence in law of this sequence. I did it using the distribution function, I have: $$F_X(x) = x \mathbb{...
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4answers
366 views

Find the probability of $a>b+c$, where $a$, $b$, $c$ are $U(0,1)$

What is the probability that $a > b + c$? $a, b, c$ are picked independently and uniformly at random from bounded interval [0,1] of $\mathbb{R}$.
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2answers
34 views

Calculate the sum of Identical uniformly distributed random variables, can't understand a specific step from the textbook

I am trying to study from a textbook and encountered the following example: No matter what I tried, I didn't understand the last step. How did the author change the limits of the integral in such way?...
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2answers
54 views

If $U$ is uniformly distributed with mean $5$ and variance $3$, what is $P(U<4)$

I'm stuck on this question, can someone help me, many thanks. If $U$ is uniformly distributed with mean $5$ and variance $3$, what is $P(U<4)$? update(this problem has been solved): I made a ...
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1answer
88 views

The pdf of sum of -log($U_i$) in which Ui is iid uniform distributed

Suppose $Ui$ is independently uniformed distributed between [0,b], $Y = -\Sigma_1^n log(U_i)$. what is the pdf of Y? I tried used characteristic function but it doesn't match each of usual ...
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1answer
31 views

Uniform and exponential distribution

Consider an experiment. The duration of the experiment has uniform distribution on $[2,6]$h. When the experiment starts, the device A turns on. This device will turn off after time,which has ...
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1answer
36 views

Hints on proving existence of $(\prod_{i=1}^{n}X_{n})^{\frac{1}{n}}$

Let $(X_{n})_{n}$ be independent random variables that are $\mathcal{U}{[1,2]}$ Prove $(\prod_{i=1}^{n}X_{n})^{\frac{1}{n}}$ exists for $n \to \infty$ and that $\exists c \in \mathbb R$ such that $(...
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0answers
74 views

“Fragmentation” of a distribution (from paper)

I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to ...
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1answer
48 views

uniform distribution on [0,1] find function

Consider $X∼unif [0,1]$. Find a function $g: \mathbb{R} \longrightarrow \mathbb{R}$, such that g(X) has pdf $f(t) = \begin{cases} {t+1}, & \text{$-1 \leq t\leq 0$} \\ {1-t}, & \text{$0<t\...
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1answer
50 views

$\ X_i $ is discrete random variable, Compute $\ \sum X_i = 97 $

Let $\ X_1, X_2, , \dots , X_{10} $ be a discrete random variable with uniform distribution between $\ 0 $ to $\ 10 $. Compute $\ P\{ \sum_{i=1}^{10} \ X_i = 97 \} $, the variables are independent. ...
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1answer
46 views

Find probability that the equation $x^2+Bx+C=0$ has 2 distinct roots.

Let $B,C$ independent random variables such that $B\sim \operatorname{exp}(\lambda),C\sim U[0,1]$. I have 2 questions about the solution: "We're looking for the probability that $\mathbb{P}(4B^2-4C&...
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0answers
48 views

What is the Probability density function of $X^2$ where X is an Uniform distribution

I'm a student and I'm studying random variables and very new to it. I was studying the Uniform distribution and in it, it calculates the Expected of $X^2$ by $$ E\left(X^2\right) = \int_{- \infty}^\...
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0answers
37 views

Binomial distribution/conjugate posterior

Let's consider some experiment with tossing a coin. NOTE: my question is given at the very last paragraph. Observation $y=0$ or $y=1$ [tails (T) or heads (H)], $p \in [0, 1]$ (probability of heads) ...
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1answer
30 views

How is the subtraction of a uniform (0, k) and its entire part distributed?

Let X be a random variable distributed as $U[0, K]$ for an integer K. Find the density function of $Y = f (x) = x- [x]$, where [x] denotes the integer part of the real number x. I think that [X] ...
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1answer
28 views

How to sample uniformly from ten urns with ten balls each

I have ten numbered urns with numbered 10 balls in each. I want to draw $n<100$ balls in a uniform distribution from all $100$ balls (the urns and all balls are distinct.) My procedure: I roll a 10-...
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0answers
110 views

probability of renewal process with uniformly inter arrival times

What is the probability the (N(t)=n) for a renewal process with interarrival time uniformly distributed on (0,1)? I thought that P(N(t)=n)== F (n)(t) − F (n+1)(t) where F (n)(t)=(t over n)/(n!). Do ...
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2answers
74 views

Pdf of $|X-Y|$ when $X,Y$ are independent Uniform $[0,a]$ variables

Need to find pdf of $ |X-Y| $ .I little confused and not getting answer after taking below limits. As it is symmetrical I have taken one part of triangle. Considering lower triangle limits I have ...
2
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2answers
35 views

product of quadratic forms of random vectors uniform on the sphere

Let $g = (g_1, ..., g_n)$ be a random vector distributed uniformly on the sphere $\{ x \in \mathbb{R}^n : \| x \|_2 = 1 \}$. Let $A, B$ be two symmetric $n \times n$ matrices. I am interested in a ...
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2answers
33 views

If $X\sim U(0,n)$, how can I show that $X-[X]\sim U(0,1)$?

If $X\sim U(0,n) ; n \in \mathbb N$ , how can I show that the distribution of $Y =X-[X]$ is $U(0,1)$? Any hint will also help me...