Cumulative Distribution Function from Probability Mass function

Both the pmf and the cdf of a discrete random variable carry the same information — the full probability distribution is determined by either one. If $X$ takes the finitely many values $x_1<x_2<\cdots<x_n$, the cdf is built directly from the pmf as the running (cumulative) sum:

$F(x)=\begin{cases}0 & x<x_1\\ f(x_1) & x_1\le x<x_2\\ f(x_1)+f(x_2) & x_2\le x<x_3\\ \ \vdots & \\ f(x_1)+\cdots+f(x_n)=1 & x\ge x_n\end{cases}$ …