Cumulative Distribution Function

We define the Cumulative Distribution Function \( F(x_0) \) as the sum of the probabilities (values of the probability density function) up-to \( X = x_0 \) for a discrete random variables.

\begin{align} F(X_0) &= \sum_{x \leq x_0} P(X = x) \label{rand-var-cumulative} \end{align}

Note: from this definition \( F(x_{\text{max}}) = 1 \).