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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21 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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44 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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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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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
87 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
24 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
62 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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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
24 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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17 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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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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61 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
81 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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33 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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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 ...
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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?
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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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130 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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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 ...
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Sort a set of arrays such that the mean-member delta of each array position is minimized

Given n number of arrays A in the format An = [xn, yn, zn] where X, Y, and Z are collections in the format X = [x1, x2, x3, ...xn], Y = [y1, y2, y3, ... yn], and Z = [z1, z2, z3, ...zn], create a ...
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32 views

Unbiased estimator for UNIF($-\theta, \theta$)

I am searching for an unbiased estimator for $\theta$ in a UNIF($-\theta, \theta$) distribution, which looks like $\hat\theta = c(X_{n:n} - X_{1:n}$). The question is to search for the c that makes ...
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1answer
67 views

Density of $ Y = X + \frac{1}{X}$ when $X\sim U(a,b)$

Let $ X $ be a continuous random variable uniformly distributed in $ \left[a, b\right] $, that is $X \sim U(a, b)$. Suppose $ 0 < a < b $. We wish to find the density function of $$ Y = X + \...
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0answers
21 views

Joint distribution of ratio of uniform distribution

If $X \sim \mathrm{Uniform}[X_{\min}, X_{\max}]$ and $Y \sim \mathrm{Uniform}[Y_{\min}, Y_{\max}]$, then what does the p.d.f of $Z = X^a/Y$ look like? I think it is a function that is segmented. ...
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1answer
35 views

Is there a notation for the distribution sampled from a set

Suppose we choose uniformly from the set S, is there a well accepted notation for this distribution? I imagine something like $X \sim U(S)$ or $X \sim Uniform(S)$? I want to use this to write that I ...
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Uniform sampling under constraints

I have a problem which is a bit weird. At some point in my program I have to sample some kind of mapping under constraints, uniformly at random. Let's say I have a certain number of unique objects, $...
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1answer
48 views

Compositions of a large integer

Let's say $n$ is the number of integers and $t$ is their sum and we want to find all the combinations of number $t$, such that there are $n$ components. See https://en.wikipedia.org/wiki/Composition_(...
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Expected remaining time for bus to come, given that already waited for 5 minutes?

There are two parts, one for when time is uniformly distributed and then an exponential distribution. Time is uniformly distributed from 0 to 10. If already waited 5 minutes, what is the expected ...
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1answer
26 views

$n^2$ times an equidistributed sequence

Suppose you have an equidistributed sequence $\{\alpha_n\}_{n=1}^\infty\subseteq(0,1)$, does $n^2\alpha_n\to\infty$ always? I don't know how to even start. Is it something like "it diverges for almost ...
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1answer
38 views

Density of Uniform distribution with respect to standard-normal distribution

How do I calculate the density function of the uniform distribution $U_{a,b}$ with respect to standard-normal distribution $N(0,1)$?
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1answer
15 views

Metric for uniformness of distribution of points in an irregular shape

I am looking for a mathematical way to check if the distribution of points inside some region (almost never a proper form) are evenly and uniformly distributed through it. Do you think this is ...
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1answer
113 views

A stick is broken and its left part is discarded.Probability that one of them $>1$ [duplicate]

A stick of length $2$ m is made of uniformly dense material. A point is chosen randomly on the stick and the stick is broken at that point. The left portion of the stick is discarded and now again ...
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1answer
26 views

Probability of $P\left(X \in \left[\frac{a + 3b}{4}, b \right] \right)$ for uniform distribution

For $X ∼ U(a,b)$, with $a,b > 0$, what is $P\left(X \in \left[\frac{a + 3b}{4}, b \right] \right)$? I do not know how to solve this. Does $$P(X \in [(a + 3b)/4, b]) = P((a + 3b)/4 < X < b)~?...
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1answer
48 views

Figuring Out Marginal Density by Looking at Plot of Joint Density

I have the below plot of the joint density of X and Y. X and Y are continuous random variables. X takes on values between 0 and 2 while Y takes on values between 0 and 1. Can someone please explain ...
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1answer
155 views

Euclidean distance of two uniform random variables

Two random variables $X$ and $Y$ are uniformly distributed, the pdfs of which are given by $f_{X}\left(x\right) = f_{Y}\left(y\right) = 1/r$. I am trying to obtain $Z = \sqrt{X^2 + Y^2}$. I tried the ...
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2answers
35 views

Joint probability density for non-identical Uniform random variables

Let $~X,~ Y~$ be uniform on $~[0, 3] × [2, 4]~$. Find $~P(X + Y ≤ 5)~$ and $~X~$ and $~Y~$ are independent. My approach: Using convolution formula. Difficulty I am facing: Understanding the limits ...
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52 views

What Distribution Summed is Uniform?

The sum of n independent uniform random variables forms a new random variable having an Irwin–Hall distribution with parameter n. (Approximately normal when n is large) $$X_i \sim \cal{U},\quad \...
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Does there exist a uniform distribution on the set of all permutations of a countably infinite set?

For a finite set $N = \{1,2,3, \ldots, n\}$, a permutation over $N$ is a bijective function $\pi: N \to N$. A uniform distribution over the set of all permutations of $N$ must assign each permutation $...
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How to predict most possible value from a set, where values don't repeat? [closed]

I have a set of readings. For example, R={69,70,73,65,170} How can I find which might be more accurate value? (It can be out of the set)
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Conditional expectation of sum of two uniform random variables

I want to compute the conditional expectation of the sum of two independent uniform random variables, with two conditions. Is there anything wrong with my approach below? Formally, let $X,Y$ both be ...
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1answer
28 views

Construct an open set containing all rational points in interval with given measure.

In a recent exam I was asked to construct an open set $M \subseteq [0,1]$ mit $\mathbb{Q} \cap [0,1] \subseteq M$ with $\lambda[M] \in (0, \frac{1}{2})$ where $\lambda$ is the uniform distribution on $...
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2answers
67 views

MLE for uniform distribution around $[-\theta,\theta]$ [duplicate]

Given $X_1,\ldots,X_n$, where $X_i\sim U(-\theta,\theta)$, what the MLE for $\theta$? Apparently the answer is $\max\{|X_1|,\dots,|X_n|\}$ but I can't figure out why. The density function is $$f(x,\...
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1answer
30 views

Distribution - Line segment quotient

Visualization of the problem Consider a line of the interval $[0,2]$ that gets divided into two parts by randomly (according to $Uniform([0,1])$ choosing one point $w$ of the interval $\Omega :=...
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1answer
29 views

Similar probabilistic results for 2 different (discrete) random variables

I have a general question, but I will motivate it with an example. Let $X_s$ be a random variable in $\lbrace1,2,\ldots\rbrace$ that follows the Zeta distribution $\zeta(s):$ $$\mathbb{P}(X_s=k)=\...
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46 views

Minimal Sufficient Statistic for $U(0, \theta)$

The definition of a Minimal Sufficient Statistic (MSS) denoted $S(X)$ is $$ \frac{L(\theta;x)}{L(\theta;x)} \text{ independent of $\theta$} \iff S(X) = S(Y), $$ assuming the densities exist and $L$ ...
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2answers
36 views

Finding CDF and PDF of $Y=20/X$ when $X$ is uniform on $[4,7]$

I have a problem where $X$ is uniform on the interval $[4,7]$ and $Y = 20/X$. I am asked to find $F_Y(y)$ and $f_Y(y)$ using the CDF and PDF. This is a uniform distribution, so it's easy enough ...
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1answer
150 views

Uniformly Distributed Random-Variable With Specific Ordering

Let $0\leq a<b$. Define the subset of $[a,b]^n$ by $$ X=\{(x_1,\cdots,x_{n-1})\mid b^{2n}\geq x_1^{2(n-1)}\geq\cdots\geq x_{n-1}^2\geq a\} $$ What is the probability that a uniformly distributed ...
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1answer
31 views

Probability (Uniform Distribution Question)

Question: A large wooden floor is laid with strips 2 inches wide with negligible space between the strips. A uniform circular disk of diameter 2.25 is dropped at random on the floor. What is the ...