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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Adjusting a set of random numbers such that they approach a uniform distribution when biased noise is added [closed]

A good random number generator, $G$ will produce a sequence of $[0, 1]$ values which are near uniformly distributed as $n$ draws goes to infinity. If I start drawing samples from the random number ...
Let $X$ and $Y$ be independent random variables with uniform density functions on $[0, 1]$. Find: $E(|X-Y|)$ $E(X)=E(Y)=1/2$, $f_{X,Y}(x,y)=1$ Integrating the region where $x > y$ and using ...
Let $X_1,\ldots,X_n$ be a a sample from a Uniform Distribution $(0,\theta)$ where $\theta > 0$ is an unknown parameter. I have found the estimator based on the sample mean \hat{\theta}=2\bar{X}\$...