The Cumulative Distribution Function: $F(x)=\mathbb{P}\{X\le x\}$

A probability mass function describes only discrete variables and a density only continuous ones, but the cumulative distribution function (c.d.f.) describes them all. For a random variable $X$ it is the function $F(x)=\mathbb{P}\{X\le x\},\qquad x\in\mathbb{R}\quad(\textbf{Definition 12.6}),$ the probability that $X$ lands at or below the level $x$. Think of $F(x)$ as the accumulated probability — the 'sum-or-area-so-far' function — sweeping a threshold from $-\infty$ rightward and recording how much probability has been collected by the time it reaches $x$. Because $F(x)$ is a probability it always lies in $[0,1]$, and because sweeping right can only add probability, $F$ can never decrease. This single object works for a die (discrete), a waiting time (continuous), or a payout that is partly a lump and partly spread out (mixed) — which is exactly why the c.d.f., rather than the p.m.f. or density, is the universal description of a random variable's distribution.

The inequality in $F(x)=\mathbb{P}\{X\le x\}$ is less-than-or-equal, so $F(x)$ is the probability of the half-line $(-\infty,x]$ including the endpoint $x$ (Remark 12.2). This is what lets the c.d.f. measure any interval: for $a<b$, $\mathbb{P}\{a<X\le b\}=\mathbb{P}\{X\le b\}-\mathbb{P}\{X\le a\}=F(b)-F(a).$ Subtracting $F(a)=\mathbb{P}\{X\le a\}$ strips off everything up to and including $a$, leaving the half-open interval $(a,b]$ — open on the left, closed on the right. For a continuous variable a single point has probability $0$, so swapping $<$ for $\le$ at an endpoint changes nothing. For a discrete variable it matters a great deal: an endpoint can carry positive mass, so $\mathbb{P}\{X\le b\}$ and $\mathbb{P}\{X<b\}$ differ by exactly $\mathbb{P}\{X=b\}$. Keeping track of which endpoints are included is therefore the whole game when computing discrete interval probabilities from $F$.

If $X$ is discrete with p.m.f. $p(k)=\mathbb{P}\{X=k\}$, then the definition becomes a sum, $F(x)=\mathbb{P}\{X\le x\}=\sum_{k\le x}p(k),$ taken over the possible values $k$ that are $\le x$ (Remark 12.3). As the threshold $x$ moves right it picks up a new term only when it crosses a possible value, so $F$ is a step function: perfectly flat between consecutive values and jumping upward by exactly $p(x)=\mathbb{P}\{X=x\}$ at each possible value $x$. The jumps add up to the total mass $1$, so $F$ climbs in a staircase from $0$ to $1$. A discrete c.d.f. is therefore not continuous — it is only right-continuous, $\lim_{s\to x^{+}}F(s)=F(x)$: at a value $x$ the function already sits at the post-jump (upper) level, because the '$\le$' includes $x$ itself. Reading the picture in reverse recovers the p.m.f.: the mass at a point is the height of the jump there, $p(x)=F(x)-\lim_{s\to x^{-}}F(s)$. Example 12.7 (the change in wealth $W$) is the model case.

For a continuous variable with density $f$, the sum becomes an integral, $F(x)=\mathbb{P}\{X\le x\}=\int_{-\infty}^{x} f(t)\,dt,$ the area under the density up to $x$; here $F$ rises smoothly instead of in jumps. For $X\sim\mathrm{Unif}[a,b]$ the density is the constant $\tfrac{1}{b-a}$ on $[a,b]$, and integrating gives the straight ramp $F(x)=\dfrac{x-a}{b-a}$ for $a\le x\le b$ (with $F=0$ below $a$ and $F=1$ above $b$), as in Example 12.8. Discrete or continuous, every c.d.f. obeys the same three laws (Proposition 12.2): it is nondecreasing with $0\le F(x)\le1$; it satisfies the limits $\lim_{x\to-\infty}F(x)=0$ (no probability has accumulated yet) and $\lim_{x\to+\infty}F(x)=1$ (all of it eventually does); and it is right-continuous. These properties are also a checklist: a function that decreases somewhere, escapes $[0,1]$, or fails the end-limits cannot be a c.d.f.