Cumulative Distribution Function or Distribution Function

There are many situations where we want the probability that $X$ is at most some value $x$, not just equal to it.

Definition 11.4 (Cumulative distribution function). For a discrete random variable $X$ taking the values $x_1<x_2<x_3<\cdots$ with pmf $f(x_i)$, the cumulative distribution function (cdf, also called the distribution function) is

While the pmf $f(x)$ is defined only at the discrete support points $x_1,x_2,\dots$, the cdf $F(x)$ is defined for every real number $x$ — it is a step function, constant between consecutive support points and jumping by $f(x_i)$ at each $x_i$. …