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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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Variance of x_i chosen from uniformly distributed hypersphere

I'm looking for an expression of the variance of a single component of a point chosen from within a uniformly distributed n-ball with radius r for any n. There are a few proofs showing that ...
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Division of two independent uniformly random variable [duplicate]

Given two independent random variable X and Y which both have uniform distribution over[0,1] I want to calculate PDF of $Z =\frac{X}{Y}$ and here is my solution: $\int_{-\infty}^{\infty}zf_X(yz)f_Y(y)...
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Variant of the Strong Law of Large Numbers

Let $X_1,X_2,\ldots$ be a i.i.d. sequence of random variables with uniform distribution on $[0,1]$, with $X_n: \Omega \to \mathbf{R}$ for each $n$. Question. Is it true that $$ \mathrm{Pr}\left(\...
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What is the distribution of the modulo of a uniformly-distributed random variable

This feels like something that's easy to answer, but maybe not. (For the record, this isn't homework from school, it's to settle an argument I'm having with a colleague.) I have a random variable $X$ ...
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Shortcut to finding the distribution of a specific random variable

Question: A dice is rolled 3 times. Let X denote the maximum of the three values rolled. What is the distribution of X (that is, P[X = x] for x = 1,2,3,4,6)? You can leave your final answer in terms ...
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Probability norm less than threshold in unit ball

From exercise 2.4 in Elements of statistical learning, studying this solution : http://tullo.ch/static/ESL-Solutions.pdf Points $x_{i}, i=1..N$ are uniformly distributed in a p-dimensional unit ball ...
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1answer
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How to compute a conditional expectation

I want to compute a conditionnal expectation, i know that $Z=(Z_1,\ldots,Z_p)'$ where $ Z_j=\Phi ^{-1}(U_j)$ with $Z \sim N(0,R(\theta))$ and $R(\theta)$ the $p \times p$ positive definite ...
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multiple distributions

Anybody can solve this question that will help me a lot. Losses due to earthquakes in a specific region are distributed uniformly in ($1MM, $5MM) and also number of earthquakes is distributed ...
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Even distribution of numbers

I'm trying to work out the most even distribution of a set of numbers across the faces of a cube. The numbers are 1-24 and I wish to place 1 number in reach corner of each face. On a standard die, ...
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1answer
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Conditioning uniform distribution on subset of support gives uniform distribution

$\newcommand{\vZ}{\boldsymbol{\mathbf{Z}}}$I am reading this paper regarding a simple proof of why rejection sampling works. I managed to understand the proof of Lemma 1, but I am struggling with the ...
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Points uniformly distributed on a circle

We select randomly two points in the circumference (with length equal to 1) of a circle. Let $X,Y$ be those points (independent and uniformly distributed) and $D$ the arc distance between them. Since ...
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Generating Standard Uniform random variable

The book I am following has a problem that states: Let $U$ be a Standard Uniform random variable. Show all the steps required to generate Then proceeds to list off questions on generating other ...
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1answer
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$E[X^k]$ and $E[(XY)^k]$

Let $X$ and $Y$ be two independent uniformly distributed random variables on $[0, 1]$. Show that $E[X^k] = \frac{1}{k+1}$ and $E[(XY)^k] = \frac{1}{(k+1)^2}$. For the first part, I used $M_{X}(t) = \...
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2answers
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Pick $2$ numbers from $[-1,1]$,what is the probability that their sum is greater than $1$?

Pick 2 numbers from $[-1,1]$, what is the probability that their sum is greater than 1? It is equal to the probability that the sum of 2 uniform random variables on $[-1,1]$ is greater than 1? so ...
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Unclear “mathematical notation” in a polynomial

Although, the Enigma here is a protocol for enhancing the privacy in blockchain; however, the question is about mathematical notation, where we want to calculate the coefficients in a polynomial. In ...
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analytical distribution of the maximum likelihood estimator for a uniform distribution

Obviously the MLE of $\theta$ for a distribution $X_1, X_2, \dots, X_n \sim Uniform(0,\theta)$ is $\hat{\theta} = max(X_1, X_2,\dots,X_n)$ Now, assume $\theta = 1$. If you take repeated samples with ...
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Acceptance and rejection sampling.

The standard logistic distribution function F given by $F(x) = 1/(1+e^{-x}) , x$ is a real number. How to generate a random sample of size 10000 using acceptance-rejection sampling using standard ...
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The Degree of the Minimal Sufficient Statistic for Θ from ~Unif(Θ-1,Θ+1)

Suppose $X_1,X_2,...,X_n$ is a random sample from the Uniform distribution over the interval $(\theta-1,\theta+1)$. By the factorization theorem, it is clear that the order statistics $Y_1=X_\left(1\...
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Interpretation of sentence to pdf

Just a quick question on interpreting what this pdf looks like: Bacteria are distributed randomly and uniformly throughout river water at the rate of $\lambda$ bacteria per unit volume. n test tubes ...
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1answer
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A doubt in finding the expected value of lifetime

Lifetime of a bulb has uniform probability distribution on (2,12). Bulb is replaced upon failure or upon reaching age 10, whichever occurs first.Find the expected value and standard deviation of age ...
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Variance of sum of two uniform RV

Let $X$ and $Y$ be two independent random variables, each uniformly distributed on $[-1,1],$ then find $\operatorname{Var}(X+Y).$ My attempt : $$\operatorname{Var}(X+Y) =\operatorname{Var}(X) + \...
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uniform distribution with interval (0,2) and sample 12

Q): Suppose that you wish to sample $12$ observations randomly from a uniform distribution on the interval $(0,2)$. An approximate value of the probability that the average of your sample will be less ...
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1answer
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Given a function that generates random numbers with uniform distribution over (0, 1) find a function to generate numbers with Bernoulli distribution.

If we have a continuous random variable $X$ with uniform distribution over $(1, 0)$ we can find functions that generate numbers with other distributions using this random variable. For example if we ...
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joint density of two sums of independent random var with common component

Suppose we have three iid draws from a uniform distribution on $[0,1]$. Call these random variables $A, B$ and $C$. Let $X=A+B$ and $Y=B+C$. I have figured out that the density of $X$ (or $Y$) is $$...
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$X_1, X_2, …, X_n \sim Exp(\lambda)$, what's the joint distribution of $X_1, X_1+X_2, …, X_1+X_2+…X_n$ and is it a uniform ordered distribution?

To elaborate on the title, here is the entire problem: Let $X_1, X_2, ..., X_n \thicksim Exp(\lambda)$ be an independent sample. What's the joint distribution of the sequence of $X_1, X_1 + X_2, ...,...
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Probability that Alice and Bob keep dating infinitely often

I solved the following problem. I would appreciate it if you can please provide feedback and let me know if I have made any mistakes. Problem statement: Online dating: On a certain day, Alice ...
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1answer
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P(X>Y) when X and Y are continuous uniform distribution

Suppose $X$ and $Y$ are continuous uniform random variables. If $X \sim U[a,b]$, $Y \sim U[c,d]$ and $[c,d] \subset [a,b]$ find the probability that a random $X$ value is greater than a random $Y$ ...
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1answer
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PDF of the product of two independent uniformly distributed random variables

Suppose that X and Y are independent U[0,1]-random variables. Find the probability density function of the product V = XY. I have seen that 𝑓(𝑧)=(−1)^(𝑛−1)log(𝑛−1)(𝑧)/(𝑛−1)! for the product of ...
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Does the Johnson–Lindenstrauss lemma require the Normal distribution?

The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly ...
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Why is quantile function of uniformly distributed random variable a random variable?

I have the quantilfe function $F^{-1}$ of a random variable which is defined as: $F^{-1}: ]0, 1[ \ni u \rightarrow F^{-1}(u) = inf\{x: F(x) \geq u\} \in \mathbb{R}$. Now I can define a new random ...
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1answer
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Relationship between cdf of normal distribution and uniform distribution defined on $[0,1]$

So the problem is, given $X\sim N(1,2^2)$, $Y=e^X$, if the cdf of standard normal distribution is $\Phi$, to show that $$\Phi\left(\frac{\ln Y-1}{2}\right)\sim U[0,1],$$ where $U[0,1]$ is the ...
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Citation for marginal of uniform hypersphere distribution

Based on some other answers (such as Marginal Distribution of Uniform Vector on Sphere), I understand that the marginal of a uniform distribution on a hypersphere is a Beta distribution. Is there a ...
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1answer
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Computing $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$ where $X_1,X_2,X_3$ are i.i.d $U(0,1)$

Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_{(1)} , X_{(2)} , X _{(3)} )$ be the corresponding order statistic. Compute $\mathbb{P}\{X_{(1)}+X_{(...
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1answer
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Compute $P(X_{(2)} ≤ 3X_{(1)})$ by using the integration technique

Suppose that $X_1,X_2,X_3.X_4$ are independent $U\in(0,1)$-distributed random variables and let $(X_{(1)}X_{(2)}X_{(3)}X_{(4)})$ be the corresponding order statistic. Compute $P(X_{(2)} ≤ 3X_{(1)})$ ...
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Conditional expectation of uniform random variable given order statistics [closed]

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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1answer
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How to Find the CDF and PDF of Uniform Distribution from Random Variable

If a random variable is defined as $Y = 3 - 2X$, how do I find the CDF and PDF of $Y$ if $X$ follows a Uniform distribution of $X \sim (-1,1)$? For CDF, am I trying to solve for $F(b) = F(-2)$ or $F(...
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Bivariate density with uniform marginals

I kindly ask for your help to solve this problem. Consider two standard uniform random variables $X_1,X_2\sim U[0,1]^2$. Then, the questions are: 1) is it possible to find the explicit form of joint ...
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Meaning of “uniformly sampling”?

Reading an article I came across the following expression: '' $\overline{X}$ is constructed by sampling uniformly along the straight line between the pair of $X$ and $\hat{X}$." I know what a ...
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1answer
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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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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 $X$. ...
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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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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-...
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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
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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
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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
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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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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 ...
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1answer
63 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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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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15 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 ...