🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

We say that a random variable $X$ is continuous if $${\displaystyle P(X=x)=0}$$ for every $x\in \mathbb{R}$

The random variable $U$ is said to have a (continuous) uniform distribution on the unit interval $[0,1]$ denoted by $U~\text{ unif }[0,1]$ iff $${\displaystyle P(U\le u)=u\forall 0\le u\le 1.}$$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function, then we say that $f$ is a density function if $f\left(x\right)\ge 0$ for any $x\in \mathbb{R}$ and $${\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)dx=1}$$

A random varaible $X$ is said to be absolutely continuousif there is a density function $f$ such that $${\displaystyle P(a\le X\le b)={\int }_a^{b}f\left(x\right)dx}$$ whenever $a\le b\in \mathbb{R}$

We say that a random variable $X$ has a density function $f$ if it is absolutely continuous using this density function.

Suppose that $Z$ is a random varaible, then we write $Z~exp\left(1\right)$ iff $${\displaystyle Z=-\ln \left(U\right)}$$ where $U~unif[0,1]$