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:

For a discrete random variable $X$, the cdf always satisfies six standing properties: (i) $0\le F(x)\le1$ for every $x$; (ii) $F$ is real-valued and non-decreasing ($x<y\Rightarrow F(x)\le F(y)$); (iii) $F$ is right-continuous ($\lim_{x\to a^+}F(x)=F(a)$); (iv) $\lim_{x\to-\infty}F(x)=0$; (v) $\lim_{x\to\infty}F(x)=1$; (vi) for $x_1<x_2$, $P(x_1<X\le x_2)=F(x_2)-F(x_1)$; and consequently $P(X>x_1)=1-F(x_1)$. …