# Generalised (general) Uniform Distribution (continuous)

I have seen the Uniform Distribution/a uniform random variable for some interval in $$\mathbb{R}$$. For example $$U(a,b)$$ has probability density function $$\frac{1}{b-a}$$ (noting this is the 'volume' of the interval)

My question : Is there a general Uniform Distribution/a general uniform random variable for a set $$\Omega$$ which is a subset of $$\mathbb{R}^{n}$$ , I assume if so the p.d.f will be the volume of $$\Omega$$ ?

If this exists could anyone tell me some of its properties like its mean, its variance, its p.d.f, /any other interesting things about it?

If this distribution (or random variable) has not been extensively studied and the above properties can not be provided, then can we make $$\Omega$$ 'nice enough' so that it does?

## 1 Answer

Its PDF would be $$1/\mathrm{Vol}\Omega$$ for $$x \in \Omega$$ and $$0$$ otherwise.

Its mean, variance, and other properties would all depend on your choice of $$\Omega$$. Without a fixed $$\Omega$$, you couldn't say anything interesting about mean, variance, etc.

This is because the PDF is still defined on all of $$R^n$$ (it's just $$0$$). It's not really uniform in that $$\Omega$$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $$\Omega \subset R^n$$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.