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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415 views

PDF of Z = XY for Jointly Uniform (X,Y) with Parabolic Region

Suppose that $(X,Y)$ is uniformly distributed on the subset of $\; \mathbb R^2$ defined by the inequalities $0 < X < 1$ and $0 < Y < X^2$. Determine the probability density function ...
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1answer
26 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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0answers
8 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
861 views

An extended warranty pays a benefit of $100$ if failure occurs between time $t = 1.5$ and $t = 8$. Find $P(w<79)$.

The time until failure, $T$, of a product is modeled by a uniform distribution on $[0, 10]$. An extended warranty pays a benefit of $100$ if failure occurs between time $t = 1.5$ and $t = 8$. The ...
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1answer
103 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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4answers
10k views

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece
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1answer
45 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 ...
3
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1answer
101 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
34 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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1answer
1k views

Uniform distribution

You arrive at a bus stop at 10 O'clock, knowing that the bus will arrive at some time uniformly distributed between $10$ and $10:30$. what is the probability that you wait longer than $10$ minutes? if ...
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2answers
47 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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1answer
814 views

Type I and type II errors

Let $X \sim uniform(0,\theta)$ we are testing $H_0: \theta = 1$ vs $H_1: \theta >1$ If we know that we reject $H_0$ if $X>0.9$ (1) find $\alpha$, the type I error (2)Suppose that $\theta=1.1$. ...
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0answers
41 views

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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1answer
33 views

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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1answer
57 views

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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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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2answers
50 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
760 views

Height of uniform distribution?

If X follows a uniform distribution in the interval [2, 7], what is the height of the probability density function (pdf) at x = 4? I'm new to Probability & Statistics and will appreciate any help!...
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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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1answer
794 views

Uniform distribution over the unit disk

Suppose that $U_1$ and $U_2$ are independent, and identically and uniformly distributed over the unit disk, i.e., for $i = 1,2$, $U_i = (X_i, Y_i)$ and the joint density is \begin{equation} f_{(X_i,...
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0answers
41 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
34 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
936 views

Uniform Distribution and First-Price Sealed Bid

For the First-Price Sealed Bid, I know that the optimal bid is $$ (n-1)/n * v_i$$ However, I am confused about a step in finding this value. We are told that there are $n$ players each with a ...
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1answer
25 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 ...
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1answer
45 views

Geometric probability of intersection of a square and a circle

In the unitary square we choose a point $(X, Y)$ with iid coordinates $U [0,1]$ and a radius $R$, independent of $(X, Y)$ and $U [0,1]$, and we draw the circle of radius $R$ with center $(X, Y)$. Find ...
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1answer
26 views

Uniform Distribution: Probability that $X$ is rational

In Rosenthal's A First Look At Rigorous Probability Theory one of the phrases about a random variable $X$ having a Uniform Distribution from $0$ to $1$ is the following: ...But now suppose we ask, ...
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1answer
32 views

prove or disprove pivot quantity uniform distribution

let $X_i\sim U(0,\theta)$ indepedent uniform variables for $1\leq i \leq 5$ prove or disprove: $\frac{X_{1}+X_{2}}{\theta}$ is pivot quantity for $\theta$. I start with $P(\frac{X_{1}+X_{2}}{\theta}...
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2answers
934 views

Maximum likelihood estimator on uniform distribution

I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me mad. Here it goes. Consider the uniform distribution on $[0,...
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0answers
15 views

Finding limits for uniform ratio distribution [duplicate]

Suppose that X∼ Uniform(a,b) and Y~ Uniform(c,d). Let Z =Y/X, I'm finding PDF Pr(Z$\leq$z). I took the region for integration as $ \frac{c}{a} \leq z \leq \frac{d}{a}$ ; $ \frac{d}{b} \leq z \leq \...
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2answers
114 views

Uniform Random Variable on $[0,1]$ and Bernoulli$(1/2)$

Let $X_1,X_2,...$ be independent, identically distributed (iid) random variables with distribution Bernoulli$(1/2)$. Define the random variable: $$Y=\sum_{n=1}^\infty\frac{X_n}{2^n}.$$ Then $Y$ is ...
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1answer
18 views

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 ...
3
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1answer
76 views

Generating vector inside a $n$-sphere

I want to generate k n-dimensional vectors which are all inside a r-radius n-sphere and the most important : I want something uniformly distributed inside the n-sphere. My initial idea is to generate ...
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1answer
117 views

Showing that if $X \sim \operatorname{Exp}(1)$, then $Y = F_X(X)$ has uniform distribution on $[0,1]$

Let $X \sim \operatorname{Exp}(1)$, and show $Y = F_X(X)$ has uniform distribution on $[0,1]$. I calculated $F_Y$, since the cumulative distribution function identifies a distribution. We have: \...
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2answers
79 views

Covariance of continuous functions, uniform and normal distribution

For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2) What is Cov(X, Y) equal to? What I tried was: E[XY] - E[X]E[Y] Where E[X] = 5.45 and E[Y] = 1.4 But for E[XY] I'm a bit clueless. I've considered:...
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1answer
54 views

Distribution function of $\sin(\pi\theta)$ when $\theta\sim U(-1,1)$

If $\theta\sim Unif[-1,1]$, then what is the CDF of $U=\sin(\pi\theta)$? Now, its easy to see that $$P_{U}(t) = P\left(\theta \leq\frac{\sin^{-1}(t)}{\pi}\right)$$ somehow the answer is equal to : ...
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1answer
49 views

Solving the quadric equation: $\frac{-U\pm \sqrt{U^2-4V}}{2}$ where $U,V$~$\mathbf{U}(-1,1)$ independently.

I'm given two uniform random variables $V,U \sim\mathbf{U}(-1,1)$. I also get the function: $$h(s)=s^2+Us+V$$ I'm interested to answer queations like for which values $h(s)$ has only: zero /one/ two ...
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21 views

Convolution of 3 uniform random variables

I really do not know how to do this. Let X have a uniform distribution on (0,100) (time to failure from 0 hours to 100 hours), avg=50. I need to determine the distribution function of 3 components. ...
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1answer
25 views

PDF of convoluted random variable conditional on another convoluted one

Suppose $V,W,$ and $X$ are mutually independent random variables. Further let $Y=V+W$ and $Z=V+X$. Is there a way to characterize the joint density $f_{Y,Z}(y,z)$ given the dependence of $Y$ and $Z$? ...
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1answer
27 views

I need to find the marginal distribution of Y from the following distributions

$f_X(x) = \frac{1}{2}e^{\frac{-x}{2}}$ and $f_{Y|X}(y|x) = I_{[0;x^2]}$ (Uniform continuous from $0$ to $x^2$). I tried finding the joint distribution by using $f(X,Y) = f(Y|X) * f(X)$ and then ...
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1answer
47 views

Probability generating function of $X\sim \text{Poisson}(\lambda)$ when $\lambda\sim U(0,2)$

The probability generating function (pgf) of $X\sim \text{Poisson}(\lambda)$ is $$G_x(t) = e^{-\lambda(1-t)}.$$ Find pgf of $X$ if $\lambda\sim \text{Unif}(0,2).$ Then find $\mathbb P(X=2).$ My ...
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1answer
1k views

Hypothesis Test with Uniform Distribution

I have a question here and not sure how to solve it, perhaps I'm overthinking it too much ! A researcher believes that the number of customers who enter a shop is uniformly distributed over 5 days ...
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1answer
18 views

showing the pdf of n-th order statistics

I am working on a mathematical stats assignment and I got stuck here. Letting $X_1, X_2, ... ,X_n$ a random sample from uniform(0,$\theta$), and Y is n-th order statistic, I need to show that the ...
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1answer
66 views

Show that $((X_1,…,X_n)|X_1+\dots+X_n=t)$, $X_i \sim Exp(1)$ is uniformly distributed

Let $X_i \sim Exp(1)$ be independent. I need to show that $((X_1,...,X_n)|X_1+\dots+X_n=t)$ is uniformly distributed over all nonnegative vectors that sum to t. What does "over all nonnegative ...
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0answers
21 views

Generating random points in a square, such that any two points must be a certain minimum distance from each other

I want to generate random points in a square, such that any two points must be a certain minimum distance from each other. I know that one way to do this would be to simply generate uniformly ...
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0answers
35 views

p-vector uniform distribution in ball and $X_i$ i.i.d exponential distribution ($\lambda$) with $ \theta = E\{ X_1 -t ∣ X_1 \gt t \}$

There's some questions: First one: Suppose $X_1, \dots, X_n$ are p-vector uniform distribution in the ball $B_\theta = \{x ∣ \Vert x \Vert \lt \theta \} ;\theta>0 $ is an unknown parameter. ...
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1answer
5k views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, $...
1
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1answer
49 views

Finding $P\left(X<\frac{3n}{2}\right)$ where $X$ is uniform on $\{n,n+1,\ldots,2n\}$ [closed]

If $X\sim \text{Uniform}\{n,n+1,\ldots,2n\}$, how can I find $P\left(X<\frac{3n}{2}\right)$ (in terms of $n$ where relevant) for both odd and even values of $n$? I got this in a test today and I ...