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 \(x=b\).