More in-depth discussion about PDFs

• Definition: A PDF $f_X$ of a random variable $X$ is a mapping from $\Omega$ to $\mathbb{R}$ with the properties:
  • Non-negativity: $f_X \geq 0$
  • Unity: Integral over sample space equals 1.
  • Meausre of a set: $\mathbb{P}[\left\{ x\in A \right\}] = \displaystyle{ \int\limits_{A} f_X\; dx }$ .
• Notice that the PMF mapped $\Omega$ to values in $[0; 1]$, but the PDF can map elements in $\Omega$ to any non negative real number.