Continuous Random Variable

probability-theory

Definition

Continuous Random Variable

A random variable $X:\Omega \to \mathbb{R}$ is called continuous if the image $X(\Omega )\subseteq \mathbb{R}$ is uncountable.

Continuous Random Variable

A random variable $X$ is called continuous if there exists a function $f:[0,\infty )$ exists, such that:

$$P(a\le X\le b)={\int }_a^{b}f(x)dx$$

Then, $f$ is called the probability density function of $X$.

Properties:

• The density $f$ is non-negative, that is $\forall x\in \mathbb{R}:f(x)\ge 0$.
  $x\notin X(\Omega )⟹f(x)=0$
• The density integrates to 1, i.e.: ${\int }_{-\infty }^{\infty }f(x)dx=1$

Note that the density is not a probability as the density can take values bigger than $1$.