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.