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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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### Confused on how to set up this uniform distribution problem

"If X is U(a,b), find the values of c and d in terms of a and b such that P(X < d) = 0.75 where a < d < b. P (c < X) = 0.9 where a < c < b" I'm confused as to what formula to ...
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### Story for the cdf $\sim x(1-x)^m$

Let $U_1,\dots,U_M$ denote $M$ uniform random variables on $[0,1]$. The PDF of $\min_i U_i$ is proportional to $(1-x)^{M-1}$. Is there such a story for the PDF that is proportional to $x(1-x)^{M-2}$?...
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### probability of a number in a range where the upper and lower bound is randomly draw from a distribution

I try to solve a probability. a and b are i.i.d. draw from a distribution, for example, 1) uniform distribution and 2) Poisson distribution in [0,1]. How to solve $Prob(a<x<b)$ where x is a ...
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### Find the CDF and PDF of $U^2$. Is the distribution of $U^2$ Uniform on $(0, 1)$?

I have the following problem: Let $U$ be a $\text{Unif}(−1,1)$ random variable on the interval $(−1,1)$. Find the CDF and PDF of $U^2$. Is the distribution of $U^2$ Uniform on $(0, 1)$? The ...
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### Proof of the Central Limit Theorem from an Infinite Number of Convolutions

In every probability book I've looked at thus far, they talk about how the sum of $n$ independent and equally distributed random variables tends towards the normal distribution as $n$ tends towards ...
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### Uniform distribution, generating function, characteristic function and

Let $$P(X = k) = \frac{4}{k(k+1)(k+2)}$$ for $k ≥ 1$. I need to show that the generating function $φ_X(u) = E(u^X)$ satisfies $φ_X(1)=1,φ′_X(1_−)=2<∞$ , but $φ′′ (1_−)=∞$. By the way, in this ...
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### Let $U_1, U_2, U_3 \sim \text{Unif}(0, 1)$. What is the CDF of $M = \max(U_1, U_2, U_3)$?

I have the following problem: Let $U_1, U_2, U_3 \sim \text{Unif}(0, 1)$, and let $L = \min(U_1, U_2, U_3)$ and $M = \max(U_1, U_2, U_3)$. Find the marginal CDF and marginal PDF of $M$. The ...
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### Where did $X | Y \sim \text{Unif}(0, 1 - Y)$ come from?

I have the following problem: Let $(X,Y)$ be a uniformly random point in the triangle in the plane with vertices $(0,0),(0,1),(1,0)$. Find the joint PDF of $X$ and $Y$, the marginal PDF of $X$, and ...
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### Step function of Poisson distributed number of uniformly distributed RVs is Poisson process

I want to proof this: Let $N,X_1,X_2,\dots$ be independent RVs, $N$ Poisson distributed and the $X_k\sim\text{Unif}([0,1])$. Then $$N_t:=\sum_{k=1}^N 1_{[0,t]}(X_k)\quad (t\in [0,1])$$ is a ...
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### What is the probability given another event is occurring?

I have Box A with $3$ red balls and $1$ blue ball. I have Box B with $1$ red ball and $4$ blue balls. I randomly take a ball from Box A and put it into Box B. I then randomly draw a ball from Box B ...
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### Find all moments of random variable when MGF not defined on open interval containing $0$.

Let $X$ be uniformly distributed on the interval $(0,1)$. Then $X$ has the moment generating function $M_x(s)=\frac{e^s-1}{s}$. I am attempting to find all moments of $X$, and I would normally expand ...
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### Jacobian method for $Z=X/Y$

I'm having trouble to solve this problem with the Jacobian method. Let $\mathbb{I}_{[0,1]^2}(x,y)$ the density of a random vector $(X,Y)$ with $X\perp Y$, i have to find the density of $Z=\frac{X}{Y}$...
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### Let $X$ be uniformly distributed in the interval $[0, 2].$ [closed]

(a) Compute $E(e^X)$ (b) Compute $E(\sin(X))$ I am confused that how to compute the interval since it did not give the formula
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### Uniform Distribution; Bus Arrival; 2 independent variables

Every evening John either visits his girlfriend who lives downtown, or visits his mother who lives uptown, but not both. In order to be completely fair, he goes to the bus stop every evening at a ...
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### Rigorous proof of CDF $F_X(X)$ is uniform

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be given. It is well known that, for continuous random variable $X(\omega)$, with CDF $F(x)$, then $F(X(\omega))$ is a uniform random variable. I want to look ...
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### Maximize the distance of pair points within a unit circle

given a unit circle and N points, how should I scatter these points inside the circle to obtain a maximum pairwise distance? i.e., the average distance (euclidean) between each pair of points will be ...
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### Generating a randomised polynomial

My question is related to this question on Cryptography SE. In the following, all the operations and polynomials are defined over a finite field of prime order, $\mathbb{F}_p$, where $p$ is a ...
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### the composition of a random variable and its cdf

Let $X$ be a continuous random variable. Let $F(t)=P(X\le t)$ be the cdf (cumulative distribution function) of $X$. Then the random variable $Y=F(X)$ takes values in the unit interval $[0,1]$. What is ...
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### Generating Uniformly Distributed Random Points in a circular region of a hyperbolic plane using this coordinate system

Let's say that in a hyperbolic plane we use a coordinate system, in which we have a u axis and a v axis that are both mutually perpendicular to each other. The coordinate lines that define u ...
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### Finding pdf of $\tan(X)$ when $X \sim U (-\pi ,\pi)$

This question is a particular case of the following: Suppose that X ∼ U ( $− π/2$ , $π/2$ ) . Find the pdf of Y = tan(X). I came up at the same solution in the following post, ignoring for a moment ...
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### Average of a set of probability measures

Given an integer $n\ge 1$, let $\textstyle \Delta:=\left\{x \in [0,1]^n: \sum_{i\le n}x_i=1\right\}$ be the $n-1$-dimensional simplex. Consider a random variable $X: \Omega\to \Delta$ (on a ...
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### Find the joint distribution of the original variables with the largest two order statistics

$k$ independent variables $X_1, X_2,...,X_k$, each has a distribtion with CDF $F_1(x), F_2(x),...,F_k(x)$. Each time we randomly take a sample of each variable (denoted as $x_1, x_2,...,x_k$) and ...
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### Expected value of general uniform order statistics

I know that if $X_k$ ~ $Unif(0,1)$ and is order statistics, then $E[X_k] = \frac{k}{n+1}$. What's $E[X_k]$ for when $X_k$ ~ $Unif(a,b)$? I think it's $a + \frac{k}{n+1}(b-a).$ Can someone confirm?