We say that a random variable $$ X$$ is continuous if $\[ P(X=x)=0 \]$ for every $$ x\in <!--\; \in \; -->\mathbb{R}$$

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

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

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

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