$\Theta \rho ϵ\eta \Pi \alpha \tau \pi$

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

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

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

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

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 $Z\sim \exp \left(1\right)$ iff $$Z=-\ln \left(U\right)$$ where $U\sim \mathrm{unif}[0,1]$