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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Expectation of the largest order statistic from uniform random variables

If $X_1, ..., X_n$ are iid random variables from the Uniform[$0,\theta$] distribution, where $\theta >0$, compute the expectation of the largest order statistic denoted $X_{(n)}$. I am looking to ...
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Find a process $\left(A_{n}\right)_{n}$ such that $\left(\sum^{n} X_{i}\right)^{2}-A_{n}$ is a martingale.

Let $\left(X_{i}\right)_i$ be a sequence of $U[-1,1]$, i.e. $f x_{i}^{(x)}=\left\{\begin{array}{ll}1 & -1<x<1 \\ 0 & \text { otherwise. }\end{array}\right.$ Assume that $\left(X_{i}\...
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1 vote
2 answers
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Expected number of rounds for a product of uniform random variables on $[1/2,3/2]$ to be for the first time below a given threshold

Starting with w=1, each time we multiply w by a number x sampled independently and uniformly from [1/2, 3/2] until it is smaller than a given value c. What's the expected number of rounds for this ...
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2 answers
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Fisher-Neyman Factorisation Theorem and sufficient statistic misunderstanding

Fisher Neyman Factorisation Theorem states that for a statistical model for $X$ with PDF / PMF $f_{\theta}$, then $T(X)$ is a sufficient statistic for $\theta$ if and only if there exists nonnegative ...
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Angle between two random unit vectors uniformly distributed

Consider $x, y \in S_{1}^{d-1}$ (the unit n-sphere in d dimensions) with $(x \cdot y)^2 = 1/d$. I need to compute the angle $\alpha$ between $x$ and $y$ for $d$ = $3$ and asymptotically for large $d$. ...
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Knowing what i is for order statistics?

is this a typo in the book or not? The question below is asking for $P(Y_2<5)$ so I thought $i=2$, but it was solved with $i=3$. Does the inequality mean that there's a plus one to $i$ ? And if it ...
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25 views

Confidence interval of a random variable

Given $Θ$ is an unknown, and $X$~$U(Θ-0.5,Θ+0.5)$ Is $[X-2,X+2]$ an 80% interval? Apparently, yes it is: $$P(X-2≤Θ≤X+2)=P(Θ-2≤X≤Θ+2)=1$$ The transition from $P(X-2≤Θ≤X+2)$ to $P(Θ-2≤X≤Θ+2)$ is ...
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X1, X2, X3 are independent and uniformly distributed over the interval (0,1). Decide the distribution of the second greatest of X1, X2, X3.

This problem is from a textbook. I have no idea where to begin. The textbook gives a lead: "The second greatest (variable) is smaller than $t$ if two or three of the $X$-variables are smaller ...
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Simple question on discrete uniform distribution

We have a random variable $X$ uniformly distributed on the set $\{1,\ldots,n\}$. Assume $s<<n$. Can anyone please advise, how to find the conditional probability $P\{X = k | X\le s \}$, where $k\...
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Ambulance problem with joint random variable

An ambulance travels back and forth, at a constant speed, along a road of length $L$. At a certain moment of time an accident occurs at a point uniformly distributed on the road. [That is, its ...
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1 vote
1 answer
43 views

Characteristic function of the uniform random variable on $[0,1]^2$

I need to calculate the characteristic function of the uniform variable on $[0,1]^2$. My idea was the following: Let $(X,Y)\sim \mathrm{Unif}\left([0,1]^2\right)$. First remark that $f_{(X,Y)}(x,y)=...
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Renewal process: find the distribution based on percentages

I was trying to solve the following renewal process problem: A policeman spends his entire day on the lookout for speeders. The policeman cruises on average approximately 10 minutes before stopping a ...
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0 answers
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Probability of an event when randomly dividing the set of items

Are we able to easily give a probability of finding a rare item if said rare items are randomly distributed amongst other items? For example, say that we have a container full of $n$ molecules, and ...
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Discrete uniform rejection region

I am currently studying for an exam. There was one problem listed that I currently am puzzled about how to approach. I would like some guidance as to what can be done in part a). I will attach a ...
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Let $Y\sim F_{\theta}$. When $g(Y, \theta)$ that does not depend on $\theta$?

Let $Y$ be a random vector in $\mathbb{R}^k$, with distribution function belonging to a family $\{F_{\theta}, \theta\in\Theta\}$ is a parametric family of distribuitiuons (e.g. normal with unknown ...
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2 answers
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Continuous Uniform distribution of 2 trees.

There are 2 trees that their heights (X,Y) distribute as follows $X$~$U(1,2)$ , $Y$~$U(1,3)$ (both continuous). X and Y are independent. The question is to find P(X>Y). My solution is as follows: $...
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2 votes
0 answers
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Explicit calculation of the arccos-1-kernel

I have given the following problem from the draft book of Francis Bach: "For $(w,b/R)$ uniform on the sphere and for the ReLU activation, compute the associated kernel as a function of the cosine ...
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Laplace's Rule of Succession and Bessel's Correction

Are applying Laplace's Rule of Succession to estimate a probability distribution from samples and applying Bessel's Correction (in reverse, perhaps) to estimate population statistics from sample ...
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Sum of a Complex Normal random variable and complex exponential of uniform random variables

I have a random variable $N \sim CN(0, \sigma^2)$. Also, consider a set of IID Uniform random variables $\Theta_i \sim Uniform (0, 2\pi]$. Consider a random variable $Y=N+A\sum_{i=1}^{k}e^{j\...
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1 answer
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What is the probability that a person will catch a bus from 8-9am under given conditions? [duplicate]

Question Find the probability that a person will catch a bus from 8 to 9 am, given that bus arrives at a random time between 8 and 9 am and waits for 11 minutes, and the person also arrives at a ...
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1 vote
1 answer
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Distribution of $k$th smallest $U_k$ where $U_k\sim \text{unif}[0, 1]$

I've seen some questions regarding this problem(order statistics). I am aware of the proof by curling the unit interval into a circle, and finding the expectation as a certain portion of the ...
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1 answer
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CDF of Uniform random variable

The problem is: $Y \sim Unif[0,\theta]$ and we define $X = -Y$. We want to find $ f_X(x)$ and $F_X(x).$ My solution: $$F_X(x) = P(X<x)=P(Y>-x)=\int_{-x}^{+\infty}{\frac{1}{\theta} d\theta}$$ But ...
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0 answers
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Rigorous proof of the Box-Muller transform method

The Box–Muller transform can be described as follows: Let $U_1$ and $U_2$ be independent uniformly distributed random variables on $(0,1)$ and \begin{align}R&:=\sqrt{-2\ln U_1};\\\Theta:=2\pi U_2.\...
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Finding $E[X^2]$ where X is uniformly distributed random variable

I'm trying to compute $E[X^2]$ where $X$ is a uniform $(0,3.3)$ random variable. This is what I tried: $E[X^2] = \int_0^{3.3}x^2 f_X (x)$ where $f_X(x) = \frac{1}{3.3}$ whenever x $\epsilon (0,3.3)$ ...
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2 answers
52 views

Covariance of min and max of n i.i.d uniform distributed random variables.

There are n independent, uniform random variables on $[a, b]$: $ξ_1...ξ_n$. How to find $ cov(min(ξ_1...ξ_n), max((ξ_1...ξ_n))$. I know that $E(min) = \frac{b+na}{n+1} $ and $E(max) =\frac{a+bn}{n+1}$
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2 votes
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two-dimensional version of "$F_X(X)$ is uniformly distributed"?

It is a well-known fact in probability theory that if $X$ is a continuous random variable and $F_X$ is a cdf of $X$, then $F_X(X)$ is uniformly distributed over $[0, 1]$. Is there a two-dimensional ...
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3 answers
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An upper bound on the probability of that the sum of k i.i.d uniform random variable is at most x?

How to prove that $\mathbb{P} (U_1 + U_2 + \dots U_k \le x) \le \dfrac{x^k}{k!}$, where $x \le 1$ and $U_1, U_2. \dots, U_k$ is $k$ i.i.d random variables $\sim Uniform[0,1]$? I have no idea except ...
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1 vote
1 answer
40 views

Mean of a random sample from a Uniform Distribution

The following question: has the following answer: The only part I don't understand is why they put E in front of 2X at the very beginning?
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2 votes
2 answers
49 views

$X,Y$ independent r.v. uniformly distributed on $[0,1]$. Find $\mathbb{P}(Y\le \frac X2)$

I'm new to these concepts, I have two approaches wich gives the same result, but I don't know if they either are both right or not. First: We must have $\mathbb{P}(Y=\frac X2)+\mathbb{P}(Y<\frac X2)...
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1 vote
2 answers
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what is the variance of difference between max and min of n i.i.d uniform variables : U(0,1)

It is an interview question: calculate the variance of difference between max and min $$variance[\max(\{X_i\}) - \min(\{X_i\})].$$ Here $\{X_i\}$ is n i.i.d uniform variables : U(0,1). I know it is ...
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1 vote
0 answers
37 views

Uniformly sampling from the cone of feasible directions in case of linear inequalities.

I'm working on an algorithm that is minimizing some loss function restricted to a polytope in $\mathbb{R}^n$: $$ \mathcal{X}= \{ x \in \mathbb{R}^n : \sum_{i=1}^n x_i \leq 1, x_i \geq 0, \ i=1,\dots,n\...
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4 votes
1 answer
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The order statistics of uniform distribution $U_{n,k_n}\rightarrow p$ a.s., when $\frac{K_n}{n} \rightarrow p$

Let $U_1,U_2,...U_n,$ be iid samples from uniform distribution $U(0,1)$. And the order Statistic: \begin{equation} U_{n,1}\leq U_{n,2}\leq ...\leq U_{n,n}, \end{equation} Given $p\in(0,1)$, if $1\leq ...
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Linear function of independent uniform distributed random variables.

Given that two independent uniform random variables X and Y, i am trying to find the function of Z = aX + bY +c. The method which I used was jacobian transformation to find the pdf. Let W = Y. Let us ...
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1 vote
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48 views

expected value for squared sum of uniform distribution

$X_1,...,X_n$ are i.i.d distributed with $X_1 \sim U(0,\theta), \theta > 0$, how does one get the expected value for $Y = (\sum X_i)^2$? For those types of questions I normally use the linearity of ...
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0 votes
0 answers
28 views

expected value of a sequence of uniform picks

This is a problem I've been nerdling on for a few days, hoping it has an elegant solution. Alas, my maths degree is now so far back in the past that I've really not got very far. I'm trying to figure ...
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9 votes
1 answer
186 views

Obtaining a tight bound for an Expectation w.r.t a uniform random variable

Let $x\in [0,1]^{n+1}$, let $t > 0$, and let $u$ be a uniform random variable over {$1,\ldots, n, n+1$}, then I want to tightly bound $$a_t(x) = E_u \left[ \mathrm{exp}\left\lbrace t\left(\frac{1}{...
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2 votes
0 answers
99 views

Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

Successively: $\pi \to e \to \log(2) \to G$ were calculated/estimated by sampling uniform distributions. Method: With a normal distribution $\pi$ can be calculated with help of the PDF (probability ...
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1 vote
1 answer
32 views

Probability task uniform distribution

currently I'm stuck with some probability task (I will provide my solution below): We have interval of 10 minutes (in other words [0; 10]). Two people ("A" and "B") come after each ...
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0 answers
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Uniform distribution over order simplex

Consider the set of all vectors $x \in [0,1]^K$ that are monotone, i.e. $0\leq x_1\leq x_2\leq ... \leq x_K\leq 1$. This set is known as orthoscheme or order simplex. Is there a formula for the ...
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3 votes
1 answer
56 views

Showing that max of uniform laws on $[0,\theta]$ is sufficient statistic with definition

Let $X_1, \cdots, X_n$ be i.i.d. $Unif(0,\theta)$ and $T = \max\{X_1,X_2,···,X_n\}$. Show that T is a sufficient statistic using the definition. So I need to show that for $t>0$, $\Bbb P(X_1 \leq ...
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  • 996
3 votes
1 answer
72 views

Find sufficient and complete statistic: Uniform distribution, $\theta \geq 1$

Take a random sample $X_1,X_2, \dots X_n$ from the distribution f$(x;\theta)=\frac{1}{\theta}$ for $0 \leq x \leq \theta$, where $\theta \geq 1$ (pay attention!). I need to find an efficient ...
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0 votes
0 answers
36 views

Joint Density Function of two uniform distributions with two discrete cases

Let there be two random variables $X_1$ and $X_2$. With probability $\gamma$, they are perfectly correlated and each distributed uniformly on the interval $[0,1]$. With probability $1-\gamma$, they ...
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3 votes
1 answer
33 views

Find Conditional expectation of uniform variables ...

Let $\xi,\eta$ be independent random variables, both with uniform distribution on $[0,2]$. Find $E[\eta^2|\xi/\eta]$. My attempt to solve the problem is in the attached file. I believe I solved it, ...
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3 votes
0 answers
54 views

Divisible approximations to the uniform distribution

As a bunch of answers on this site have pointed out: Can sum of two random variables be uniformly distributed, https://mathoverflow.net/q/228014/5429, the Uniform distribution is not divisible. That ...
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1 vote
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how to compute E[X] given X+Y>1, X, Y iid ~ uniform (0, 1)

Given $X, Y$ iid $\sim \operatorname{uniform} (0, 1)$ one way is $F(x\mid X+Y>1) = P(X<x\mid X+Y>1) = \frac{ P(1-Y<X<x)}{ P(X+Y>1)} = 2 (x-1+y),$ this is incorrect. or, $E[X\mid X+...
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3 votes
1 answer
93 views

Proving an inequality regarding the tenth moment

Let $Y$ be a random variable with $\mathbb{E}(Y \mid X) = X$, where $X \sim \mathrm{U}[0,1]$. Prove that $\mathbb{E}(Y^{10}) \geq \frac{1}{11}$. I tried using Jensen's inequality and found out that $\...
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  • 83
0 votes
0 answers
32 views

Chi-square testing if distribution is uniform

I have a question around the Chi-Square test which I'm trying to determine if a given set of values has uniform distribution. I care about whether or not it can be shown with statistical significace ...
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0 answers
47 views

Find posterior distribution and Bayes estimator

Suppose that an observation $x \in (-1,1)$ comes from a sample model with a parameter $\theta$, with density function: $$ f(x\mid\theta) = \begin{cases} \theta\ if -1 < x < 0\\ 1 - \...
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10 views

distribution function of a random variable uniformly distributed on a set

How can I find the distribution function of a random variable uniformly distributed on a set $$\text{(}0, 1\text{)}\;\cup \; \text{(}-2, -1\text{)}\; \cup\text{(}3, 5\text{)}$$
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2 answers
61 views

Finding the probability for the sum of 2 variables [closed]

I’ve been given the joint density function: f$_X$$_,$$_Y$(x,y)=C when (X,Y) is uniform over [-1,1]$^2$. I’ve been tasked with finding P{|2X+Y|$\le$1} and P{X=Y} however I’m stuck in my question, I’ve ...
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