Retrieving PDF from CDF

Now is when the fundamental theorem of Calculus (discussed in the first chapter) comes into play.

• Theorem: The probability density function (PDF) and CDF satisfy:
  • \(f_X(x) = \dfrac{d F_X (x)}{dx}\), is \(F_X\) is differentiable at \(x\) .
  • \(f_X(x_0) = \mathbb{P}[X=x_0]\delta(x - x_0)\), is \(F_X\) is not differentiable at \(x=x_0\) .