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.

2209 questions
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How to transform gaussian(normal) distribution to uniform distribution?

I have gaussian distributed numbers with mean 0 and variance 0.2. And I want to transform this distribution to uniform distribution [-3 3]. How can I transform gaussian distribution numbers to uniform distribution?
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2 answers

Why are all subset sizes equiprobable if elements are independently included with probability uniform over $[0,1]$?

A probability $p$ is chosen uniformly randomly from $[0,1]$, and then a subset of a set of $n$ elements is formed by including each element independently with probability $p$. In answering Probability of an event if r out of n events were true. I…
joriki
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Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. Let $u$ be an independent of $\{ \xi _a \}_{a \in…
11
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2 answers

Prove there exists no uniform distribution on a countable and infinite set.

Can anyone help me with this problem, I can't figure out how to solve it... Let $X$ be a random variable which can take an infinite and countable set of values. Prove that $X$ cannot be uniformly distributed among these values. Thanks!
user52516
11
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1 answer

Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed random points inside the rectangle, and why?
10
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2 answers

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
9
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2 answers

Draws from the uniform distribution are taken until the sum exceeds 1. What is the expected value of the final draw?

I was thinking about this question after a related problem: what's the expected number of draws for the sum to exceed 1? For that problem, the answer is known and is a surprising result http://mathworld.wolfram.com/UniformSumDistribution.html My…
Presh
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9
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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}{n}\sum_{i\neq u}x_i - x_u \right) \right\rbrace…
9
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Can the sum of two independent r.v.'s with convex support be uniformly distributed?

Is it possible to prove that the sum of two independent r.v.'s $X$ and $Y$ with convex support cannot be uniformly distributed on an interval $[a,b]$, with $a < b$? (Let us rule out the trivial case where $X$ is degenerate and $Y$ is uniform.)…
mlc
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9
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Normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = \left(\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{X_n}{Z}\right) $$ is a uniform random vector on…
9
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2 answers

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution function of $S=X+Y$.
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distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How can I continue from here?
8
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2 answers

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process $\left\{\tilde{B}\left(t\right): t \geq 0\right\}$…
8
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3 answers

X,Y ~ Unif(0,1) not necessarily independent, can P(X+Y>1)>1/2?

My set up is the following: $X,Y \sim \text{Unif}(0,1)$ but their joint distribution is not constrained. My question is whether there exists a joint dependence between them (that preserves the marginals) such that $\operatorname{Prob}(X+Y>1)>1/2$. I…
8
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The expectation of $e^X \left(1-(1-e^{-X}\right)^n)$ when $X$ has Exponential Distribution

To my surprise, I was able to evaluate the following expression in Mathematica: $$E\left[e^X \left(1-(1-e^{-X}\right)^n) \right] = \frac{y}{y-1} \left(1-\frac{1}{\binom{n+y-1}{y-1}}\right)\quad X\sim\text{Exp}(y)$$ with the right hand side being…