Tail Or Cumulative Probabilities

1. Core Concept Overview & Intuition

A tail probability collects probability mass on one side of a threshold. For a non-negative integer-valued random variable, the upper tail records the chance of exceeding a value, while the lower cumulative probability records the chance of being at or below it. Their generating functions can be expressed using the probability generating function (p.g.f.) of the original variable.

2. Mathematical Formalization & Core Definitions

Let $p_k=\Pr(X=k)$ and let

$$
P(s)=\sum_{k=0}^{\infty}p_ks^k.
$$

The upper-tail probability and its generating function are

$$
q_k=\Pr(X>k)=\sum_{j=k+1}^{\infty}p_j,
\qquad Q(s)=\sum_{k=0}^{\infty}q_ks^k.
$$

Because $q_k$ is a tail sequence, $Q(s)$ is a generating function; it is not generally a p.g.f. The lower cumulative probability is

$$
r_k=\Pr(X\leq k)=\sum_{j=0}^{k}p_j,
\qquad R(s)=\sum_{k=0}^{\infty}r_ks^k.
$$