For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

# Questions tagged [uniform-distribution]

2209 questions

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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?

Theodore Murdock

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### When is the sum of two uniform random variables uniform?

Suppose that $X$ and $Y$, two random variables, are both uniformly distributed over $[0,1]$. Let $Z=\frac{1}{2}X+\frac{1}{2}Y$.
I know that in general, $Z$ is not uniform. For instance, $Z$ is not uniform if $X$ and $Y$ are independent.
On the other…

Yi-Hsuan Lin

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### Binomial distribution with random parameter uniformly distributed

I have a problem with the following exercise from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001 (page 155, ex. 6):
Let $X$ have the binomial distribution bin($n$, $U$), where $U$ is uniform…

user52354534

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### Improper Uniform Distribution on $\mathbb{R}$

It is not possible to write a function representing the uniform distribution, say $\mathcal{U}$, on the real line, $\mathbb{R}$ (in a Bayesian context, this is an improper prior). My question is: is it possible to write a generalized function (aka,…

PseudoRandom

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### Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$

Let $X_{1},\ldots,X_{n}$ be i.i.d. $U[\alpha,\beta]$ r.v.s., and let $X_{(1)}$ denote the $\min$, and $X_{(n)}$ the $\max$. Show that
$$
\mathbb{E}\left(\overline{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}.
$$
I know that…

GurrVasa

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### Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the limit.
I believe the limit is $0$. I tried to…

Landon Carter

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### Density of sum of two uniform random variables

I have two uniform random varibles. $X$ is uniform over $[\frac{1}{2},1]$ and $Y$ is uniform over $[0,1]$. I want to find the density funciton for $Z=X+Y$.
There are many solutions to this on this site where the two variables have the same range…

Gastra

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### Joint density problem. Two uniform distributions

This is the problem:
An insurer estimates that Smith's time until death is uniformly distributed on the interval [0,5], and Jone's time until death also uniformly distributed on the interval [0,10]. The insurer assumes the two times of death are…

user153729

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### Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting:
http://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution
On the Wikipedia article above they derive…

user77404

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### Distribution of $\max(X_i)\mid\min(X_i)$ when $X_i$ are i.i.d uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) \mid \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$
I agree that the probability density function of $\max(x_i) \mid \min(x_i)$ must…

jrand

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### If $X$ is Gaussian, prove that $X-\lfloor X \rfloor \sim U(0,1)$ as its variance becomes large

I have a normal distributed random variable, $X$ with mean $\mu$ and standard deviation, $\sigma$. I don't believe it matters, but this distribution was obtained as a result of summing a large number of independent, identically distributed random…

Rohit Pandey

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### Find a minimal sufficient statistic for $U(\theta,\theta+c)$, where $(\theta,c)$ unknown.

Suppose $X_1,\cdots,X_n$ are $i.i.d$ from a distribution with p.d.f
$$\delta_{(\theta,c)}(x)=\frac{1}{c}\mathbb{1}_{(x\in[\theta,\theta+c])},$$
where $\theta\in\mathbb{R}$ and $c\in\mathbb{R}^+$ unknown.
Find a minimal sufficient statistic for…

Tan

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### MLE for Uniform $(0,\theta)$

I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$.
I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelihood function.
What is confusing me is that if a function…

hyg17

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### Sum of 3 uniform random variables is a constant

Give a construction of three random variables $X,Y,Z$ that are each
uniform on $(0,1)$ but $X+Y+Z$ is a constant.
Is the following argument correct? We first consider every number in $(0,1)$ in ternary and for the $n$-th digit, randomly assign…

iYOA

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### Expected determinant of a random symmetric matrix

The three distinct entries of a $2 \times 2$ symmetric matrix are drawn from the uniform distribution over $[-60, 60]$. What is the expected determinant of the matrix?
I assume it is $0$ but I am not good at proving it efficiently. Thanks.

Datmach

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