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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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How to transform a $U(0,1)$ variable to produce a Poisson variable?

Suppose $ X $ is a uniformly distribution over $(0,1)$. How to find transformations $Y=g(X)$ to produce random variables with the Poisson distribution?
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1answer
23 views

Finding density function of random variable

Choose an uniformly distributed random variable $U$ on the unit interval $[0,1]$. Then, what is the probability density function of $Y= \ln(U+ 1)$? I know the density function is the derivative of ...
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1answer
22 views

How can calculate conditional pdf of Y when you dont know about f(y)

X is a uniform distribution on the interval (0,1). Y is a also uniform distribution on the interval (0,x). Its the only information that I could know. Then how can I calculate p(Y|x)? If you teach me, ...
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According to the uniform distribution U(0,θ), how to obtain the suitably normalized limit distribution for ((n+1)/n) X(n) ) [on hold]

I was thinking of doing it by means of the maximum likelihood estimator but how can I apply it to the case in which I get (n + 1) / n?
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28 views

Hard demonstration - brilliant minds. I came up with the idea of ​obtaining the limit distribution of an estimator, [on hold]

I do not know exactly how to get there, I thought to use the estimate of maximum likelihood but I'm not sure... If $X_1,\ldots , X_n$ are i.i.d. according to the uniform distribution ${\cal U} (0, \...
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23 views

understanding the conditional entropy in the case of having uniform distribution?

Would you please help me to understand the conditional entropy in this example which I got stuck in? The example Considers 4 uniformly popular binary vectors, for example; {f1,f2,f3,f4} each with ...
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1answer
23 views

Uniform distributed success probability for a coin

$n\in \Bbb N$. Let $X_1 \sim \text{Uni}_{(0,1)}$ and $X_2 \sim \text{Bin}_{n, X_1}$ conditional on $X_1$. I want to find the distribution function of the law of $X_1$ given $X_2 = k$, i.e. $\Bbb P (...
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4answers
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Frog makes two jumps (uniform distribution)

I received this problem on my exam, although I thought I answered it right, it was marked as wrong. There is a frog on a line. The frog starts from a point 0 and makes two successive jumps: first ...
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25 views

P/1 Actuary Question: Expected value for continuous uniform distribution.

A question reads: A loss random variable has a continuous uniform distribution on the interval $(0, 100)$. A insurance policy on the loss pays the full amount of the loss if the loss is less than or ...
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2answers
30 views

Probability mass function of $ \min(X, Y)$ where $X,Y$ are i.i.d discrete uniform

Let $X$ and $Y$ be two discrete uniform i.i.d random variables distributed over $\{0, 1, 2,\ldots, N\}$. Find the pmf of $Z = \min(X, Y)$. From what I understand, I have to find the joint $pmf$ first,...
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58 views

Coverage probability for Uniform$(0, \theta)$

Let $X_1 \dots X_n$ denote a random sample from a uniform $(0, \theta$) distribution. PROBLEM: Compute the coverage probability for the CI: $$\left(\frac{X_{(n)}}{0.95}, \frac{X_{(n)}}{0.25}\right)...
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What is P($X_{1}>X_{2}>…X_{n-1}>X_{n}$)?

Given $X_{i}$, $1 \leq i\leq n$, are independent random variables with uniform distributions on $[0, 1]$, what is P($X_{1}>X_{2}>...X_{n-1}>X_{n}$)? I thought it would be something along the ...
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0answers
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JPDF for uniform-distribution

you are given that the joint distribution of X and Y is uniform on the region defined by the conditions: $0<x<1$, $x<y<x+1$. Find the correlation coefficient of X and Y My problem with ...
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Alternate characterization of a spatial Poisson point process

Would I be correct to assess that a spatial Poisson point process on some compact, say the $d$-dimensional sphere, can be simulated by first choosing some $n \sim \mathrm{Poisson} (\beta)$ number of ...
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25 views

Probability of uniform distribution with poisson process

Men and women arrive at a store according to independent Poisson processes with hourly rates $\lambda_M$ = 3 and $\lambda_F$ = 4, respectively. Men shop for a time that is uniformly distributed on [0, ...
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34 views

There are $45$ boxes, each of which contains $N$ weapons. If he distributes all the weapons evenly to his $2026$ soldiers,

There are $45$ boxes, each of which contains $N$ weapons. If he distributes all the weapons evenly to his $2026$ soldiers, he would have $2016$ weapons left over. What is the smallest positive value ...
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Consistency of MLE for uniform distribution $U[\theta,1/\theta]$ where $0<\theta<1$

Since the likelihood function of $(X_1,\ldots,X_n)$ is $$L(\theta)=\left(\frac{\theta}{1-\theta^2}\right)^n I(\theta < x_{(1)}) I(\theta < 1/x_{(n)})$$ So the MLE of $\theta$ is $$\hat{\theta}=...
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1answer
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Find probability of combination of two uniform variates [closed]

$X \sim R(0, 2)$, $Y \sim R(0, 5)$, X and Y are independent, I need to find $P(|X-Y| \leq 1)$
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1answer
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How to find critical value in uniform distribution [closed]

How do you get critical values from uniform distribution p.d.f?? this is the question I got from my teacher enter image description here
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2answers
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Probability distribution of a uniform distribution to the third power

I have to find a explicit form of probability distribution of $X^3$, if $X \ \mathtt{\sim} \ U[a, b], \ -\infty < a < b < \infty$. So far I've succesfully done a simpler version, when it's $\...
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1answer
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Expectation of -log(U)

Let $U$ be a uniform distribution on $[0,1]$ 1) Find the distribution function of $V = -log(U)$ (where log is the natural log) 2) Find $E(V)$ What I got: 1) $F_V(x) = P(V<x) = P(-log(U) < x) ...
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1answer
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Universality of uniform: plugging a random variable into its CDF?

Consider the universality theorem of the uniform distribution. One way to formulate it is the following: Let $F:\mathbb{R}\rightarrow [0,1]$ be a right continuous, increasing function. Then, if $X\sim ...
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On the sum of two random variables with uniform joint distribution on a parallelogram

Given $(X ,Y)$ uniformly distributed over the parallelogram with vertices $(-1,0)$, $(1,0)$, $(2,1)$, $(0,1)$, I'm intrested in the distribution of $Z = X+Y$. Solution The contours of (X, Y) look ...
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2answers
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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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2answers
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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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2answers
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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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2answers
25 views

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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1answer
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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
28 views

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
22 views

$E[X^k]$ 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
203 views

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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0answers
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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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1answer
44 views

Degree of the minimal sufficient statistic for $\theta$ in $U(\theta-1,\theta+1)$ distribution

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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1answer
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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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1answer
84 views

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
29 views

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 ...