# Tagged Questions

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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### Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
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### How To prove Any Change to $v=a\cdot y + b$ maks $y=(a)^{-1}(v-b)$ Uni. random value

This question may seem to be related to Probability and Data Integrity but mine is much simpler and consideres a DIFFERENT problem. Let a finite field be $\mathbb{Z}_p$, where $p$ is a prime number. ...
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### Show that there is no discrete uniform distribution on N.

This is a homework question I got. I'm not entirely sure what it is asking. Can someone please clarify/get me on the right track?
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### Why $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent if $X_1$ and $X_2$ are i.i.d. exponential?

How to show that $2\frac{X_1}{X_1+X_2}-1$ and $X_1+X_2$ are independent, if $X_1$ and $X_2$ are i.i.d. exponential with mean $1$? Is there a simple way to see this?
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### Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $F(x) = P[X\le x]$ is continuous in x, show that $Y=F(X)$ is measurable and that $Y$ has a uniform distribution $P[Y\le y] = y, \; 0\le y \le 1$ My first question is about notation. What ...
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### equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
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### Rao-Blackwell Uniform Distribution

I am having a bit of an argument with my study group about a Rao-Blackwell problem that we have for our statistical theory class. The problem goes like this: Let X~U(0,$\theta$), and suppose we have ...
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### Uniform distribution, Expected value and standard deviation for proportion of observations in a subintervall

$X\sim U(0,1)$. Divide the interval [0,1] into k equal subintervals. Then $X_1$=the number of observtions on the first interval. Define the new variable $Y_1=X_1/n$, where n is the number of ...
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Suppose a set of $m$ integers from $0$ to $n-1$. The integers are uniformly distributed and unique in the set ($n \gg m$). Then, put all the integers into a list an sort that list: $$x_0 < x_1 < ... 1answer 28 views ### Finding a uniform distribution on the output of a multivariable function Suppose we have a non-invertible continuous function that maps from some continuous interval {I}^n to \mathbb{R} with n \ge 1. To take an example, let f(a,b,c) = a \cdot e^{-bc} - b \cdot e^{\... 1answer 50 views ### Proving that a statistic is not sufficient (uniform case). Let X=(X_1,...,X_n) be i.i.d. U(0,\theta). How to show that$$\frac{2}{n}\sum_{i=1}^{n}X_i$$is not a sufficient statistic? I have already proven that \max_{i=1,...,n}X_i is a sufficient ... 1answer 777 views ### Sufficient statistic for uniform distribution Given random sample \left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\}  from  U(0,\theta). Let {Y}_{i} be the order statistics. Then the sufficient statistic for \theta is { Y }_{ n }... 1answer 52 views ### Why is X/\|X\|_2 uniformly distributed on a unit sphere when X is n-dimensional standard gaussian vector? In the proving the above, I see that since X is multivariate gaussian then for any orthogonal matrix Q we have that QX is standard multivariate gaussian. Then I somehow reasoned that Y=X/\|X\|... 3answers 54 views ### Find the joint distribution of the continuous order statistics? Take n independent variables, {X_1, X_2,\dots, X_n}, which are uniformly distributed over the interval (0,1). Then, introduce the variable M=\min(X_1, X_2,\dots, X_n) and the variable N=\max(... 2answers 207 views ### Probability of maximum of two uniform random variables The random variables X and Y are independent, each with the uniform distribution on [−1, 1]. Find:$$P[\max (X,Y) >0.5] Apparently there is an easy approach without integration, but I am ...
Question: Suppose there are Y types of balls in a bucket, which are normally distributed and independent. Hence the probability of picking one type out is $\frac{1}{Y}$. Let $x$ be the number of ...
$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...