Cumulative Distribution Function (c. d. f.)

The p.m.f. fully specifies a discrete random variable's probability distribution, but it is sometimes more convenient to work with a running total instead: the cumulative distribution function (c.d.f.) of a discrete random variable $X$, denoted $F$, is defined by $F(x) = P[X \le x] = \sum_{x_i \le x} P[X = x_i] = \sum_{x_i \le x} p_i = \sum_{x_i \le x} f(x_i)$, where $f$ is the p.m.f. of $X$. In words, $F(x)$ adds up the p.m.f. over every possible value that does not exceed $x$.

Reusing the four-coin-toss experiment ($X$ = number of heads in 4 tosses), Table 7.4 places $f(x) = P[X=x]$ and $F(x) = P[X \le x]$ side by side: $F(0) = 1/16$; $F(1) = 1/16+1/4 = 5/16$; $F(2) = 5/16+3/8 = 11/16$; $F(3) = 11/16+1/4 = 15/16$; $F(4) = 15/16+1/16 = 1$. A second illustration returns to the coin-tossed-till-first-head experiment; here $X$ ranges over every positive integer, and Table 7.5 shows the pattern continuing indefinitely, with $f(x)$ halving at each step ($1/2, 1/4, 1/8, 1/16, 1/32, \ldots$) and $F(x)$ climbing towards, but never in finitely many steps reaching, 1 ($1/2, 3/4, 7/8, 15/16, 31/32, \ldots$).

Because a discrete random variable jumps only at its possible values and is flat everywhere else, its c.d.f. $F(x)$ is always a non-decreasing step function — a fact contrasted again for the continuous case in section 7.4.2, where the corresponding $F(x)$ turns out to be a non-decreasing continuous function instead. …