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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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7 views

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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20 views

Expected value of the $k$'th order statistic given that it's smaller than $\tau$

Let $X_1,\ldots,X_n\sim U[0,1]$ be i.i.d. uniform random variables and let $X_{(k)} $ denote the $k$'th smallest variable. Given some $\tau\in(0,1)$, what is $$\mathbb E[X_{(k)}\mid X_{(k)}\le \tau]?$...
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11 views

Uniform prior distribution on log scale

Can anyone please suggest me a distribution whose $log$ transformation is uniform and it should be a well known distribution? I am not sure if it exists. I know that if we consider a uniform ...
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0answers
14 views

Components of an n-dimensional random variable from a uniform distribution are independent

For a random variable $X \in \mathcal{R}^n$ from a uniform distribution, are the components independent? i.e, $Pr(X_1,X_2,...,X_n) = Pr(X_1)Pr(X_2)....Pr(X_n)$ If so, does it depend on the uniform ...
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1answer
25 views

Find density of $(X,Y)$ when $X= R \cos{(2\pi U)}$ and $Y=R \sin{(2\pi U)}$

Let $R$ and $U$ be independent random variables where $R$ has density $f_{R}(r)=2 r\cdot1_{[0,1]}(r)$ and $U$ is the uniform distribution on $[0,1]$. Furthermore let $X= R \cos{(2\pi U)}$ and $Y=R \...
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0answers
12 views

How concentrated is the $t^{th}$ smallest discrete uniform order statistic?

Let $n,z,t$ be positive integers and let $X_1,\ldots,X_{z\cdot t}$ be i.i.d. random variables that are uniformly distributed over $\{0,\ldots,n\}$. Let $X_{(t)}$ denote the $t^{th}$ smallest ...
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1answer
23 views

Find CDF of random variable Y=X(2-X)

Let $X$ is uniformly distributed over the interval $[0,2]$.I need to find CDF of random variable $Y=X(2-X)$. My solution: $$\begin{align}F_Y(t)&=P(Y \le t) \\[1ex]&=P(X(2-X)\le t) \\[1ex]&...
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9 views

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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22 views

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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1answer
29 views

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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1answer
31 views

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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1answer
27 views

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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2answers
29 views

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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1answer
22 views

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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2answers
31 views

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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1answer
45 views

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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1answer
21 views

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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1answer
44 views

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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2answers
27 views

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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1answer
123 views

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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1answer
43 views

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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0answers
28 views

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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0answers
26 views

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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1answer
13 views

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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1answer
27 views

How to find the cube of a uniform distribution?

I came across this question in a quiz and I was not sure on how to do it since my lecturer didn't clearly teach this. Can anyone assist me with this: Let $X$ ~ $U(0,3)$, find the density $f(u)$ for $...
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2answers
49 views

2D LOTUS: joint PDF on unit square $\{ (x, y) : x, y \in [0, 1] \}$

My textbook, Introduction to Probability by Blitzstein and Hwang, presents the following example: Example 7.2.2 (Expected distance between two Uniforms). Let $X$ and $Y$ be i.i.d. Unif$(0, 1)$ r.v....
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1answer
20 views

For a, b in (−∞, ∞), a < b, show that Y = a+(b−a)X has a uniform distribution over [a, b]. [closed]

Let X be uniformly distributed over the interval [0, 1] For a, b in (−∞, ∞), a < b, show that Y = a+(b−a)X has a uniform distribution over [a, b]. How would you approach this problem? Uniform ...
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2answers
36 views

What is $P(X_1>X_2+X_3)$ if $X_i$s are i.i.d uniform $(0,1)$ variables?

Let $X_1,X_2$ and $X_3$ be iid $U(0,1)$ random variables. Then $P(X_1>X_2+X_3)$ equals? What I think may be correct $$P(X_1>X_2+X_3) =\int\int \int _{x_1>x_2+x_3} 1.1.1\,dx_1\,dx_2\,dx_3=\...
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1answer
20 views

Find the a posteriori probability? (Ch-4,Exercise-21, Probability, Random Variables and Stochastic Processes-Papoulis)

The probability of heads of a random coin is a random variable p uniform in the interval (0, 1). (a) Find P{O.3 <= P <= O.7}. (b) The coin is tossed 10 times and heads shows 6 times. Find the a ...
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1answer
43 views

Is $3-4U$ equivalent to $4(1-U)$?

If $U$ is defined to be the uniform distribution on $(0,1)$, is it true that $3-4U\sim4(1-U)$?
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1answer
19 views

Is this an equivalent uniform distribution?

Suppose $U$ is uniformly distributed on $(0,1)$ then it is true that $X=\ln(1-U)\equiv \ln(U)$. This makes sense because $1-U\sim U$, and thus $X$ will have the same distribution. Now, suppose $c\in(0,...
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1answer
19 views

Variance of (X + Y)^2 where both X and Y are uniformly distributed between 0 and 1?

I can't seem to figure out how to solve that. If X and Y are two independent random variables X and Y sampled uniformly from the [0, 1], then what is the variance of $(X+Y)^2$ ? I know that $Var((X+Y)...
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2answers
83 views

Expected number of regions with $n$ random lines in a circle

There are $n$ random lines drawn in a circle, defined by endpoints being uniform on circle. I am trying to figure out the expected number of regions separated by $n$ lines. I know $f(0)=1$, $f(1)=2$ ...
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2answers
47 views

Distribution of $Y=X(1-X)$ when $X\sim U(0,L)$

I was wondering if someone could provide a hint of how to solve the following problem. I am working on some of my own research and can't seem to remember how to solve a problem like this: Let $X\...
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1answer
23 views

conditional distribution of sum of order statistic

When I read Ross's book "Statistic Process" , I find the lemma, but I cannot prove it. The lemma states that Let $Y_1, \cdots, Y_n$ be iid nonnegative random variables then $E[Y_1+\cdots + Y_k| Y_1+\...
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1answer
60 views

The density function of the sum of n independent uniformly distributed random variable on $(-1,1)$ is supported on $[-n,n]$

This question arose from the statement after Exercise 3.3.6 of Durrett's probability. Exercise 3.3.6: Show that if $X_{1}, \cdots, X_{n}$ are independent uniformly distributed on $(-1,1)$, then for ...
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0answers
54 views

Universality of the uniform in the context of the Rayleigh distribution

I am currently exploring a theorem called universality of the uniform: Let $F$ be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that ...
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1answer
23 views

Let $X$ be an r.v. with CDF $F$. Then $F(X) \sim \text{Unif}(0,1)$?

I recently encountered a theorem called universality of the uniform: Let $F$ be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that ...
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1answer
13 views

Uniform distributions: location-scale transformation

My textbook, Introduction to Probability, first edition, by Blitzstein and Hwang, says the following: In a location-scale transformation, starting with $X \sim \text{Unif}(a, b)$ and transforming ...
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1answer
42 views

Showing independence of ratio of ordered statistics for uniform distribution

From An Intermediate Course in Probability by Allan Gut Suppose that $X\in U(0,1)$. Let $X_{(1)},X_{(2)},\ldots,X_{(n)}$ be the order variables corresponding to a sample of $n$ independent ...
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1answer
50 views

Let $U_1,U_2,…$ be a sequence of independent uniform $(0, 1) $random variables, question about $\Pr(N > n)$

Let $U_1,U_2,...$ be a sequence of independent uniform $(0, 1) $random variables and let $$N:=\min\{n\geq 2: U_n>U_{n-1}\}$$ $$M:=\min\{n\geq 2: U_{1}+\cdots+U_n>1\}$$ Show that ...
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1answer
29 views

What is the distribution when you add exponential rvs to uniform rvs and count?

Let $X_1, \dots, X_c$ be independent uniform random variables on $[0,1]$. We know the number of random variables whose value will fall in the range $[0,1]$ will always be exactly $c$. Now define $...
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2answers
67 views

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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1answer
70 views

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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0answers
31 views

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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0answers
28 views

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 ...
3
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0answers
22 views

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?
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59 views

Product of two discrete uniform distributions

Let $a$ and $b$ be positive integers with $b > a$. Let $A$ and $B$ be two discrete uniform distributions over the integer intervals $[[1, a]]$ and $[[1, b]]$ respectively. Then, given any $t \in [...
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1answer
102 views

Twist on classic interview question. $P(X > 3Y)$ where $X,Y$ are uniform random variables

So there's this very classic probability question that says: Given$ X, Y$ two INDEPENDENT uniform random variables in $[-1,1]$, what is $P(X > 3Y)$? Of course, there are alterations where the ...
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3answers
21 views

How do you compute uniform distribution with only mean and proportion given?

Machine A produces mints that have a label weight of 50g and is believed the weights of the weight is uniformly distributed, with a mean of 51.5g and 70% of them less than 52.5g. What's the ...