Properties of CDF

• Theorem: Let $X$ be a random variable (continuous or discrete), the its CDF satisfies the following properties:
  1. The CDF is a non-decreasing.
  2. The maximum of the CDF is when $x=\infty$: $F_X (\infty) = 1$.
  3. The minimum of the CDF is when $x=-\infty$: $F_X (\infty) = 0$.

 

• Definition: A function $F_X(x)$ is said to be:

  • Left-continuous at $x=b$ if $F_X(x) = F_X(b^-) = \lim\limits_{x' \to b^-} F_X(x')$
  • Right-continuous at $x=b$ if $F_X(x) = F_X(b^+) = \lim\limits_{x' \to b^+} F_X(x')$
  • Continuous at $X=b$ if it is both left-continuous and right-continuous.

• Theorem: The CDF of any random variable (discrete or continuous) is always right-continuous.

• Theorem: For any random variable $X$ (discrete or continuous):

  • $\mathbb{P}[X=b] = 0,$ if $F_X$ is continous at $x=b$.
  • $\mathbb{P}[X=b] = F_X(b) - F_X(b^-),$ if $F_X$ is discontinous at $x=b$.