Cumulative Distribution Function¶

Definition¶

For a real-valued random variable $X$ , its cumulative distribution function is

F_X(x)=\mathbb{P}(X\le x),\qquad x\in\mathbb{R}.

It is nondecreasing and right-continuous, with $\lim_{x\to-\infty}F_X(x)=0$ and $\lim_{x\to\infty}F_X(x)=1$ $1$ . These limits describe the notation $F_X(-\infty)=0$ and $F_X(\infty)=1$ $1$ .

Interpretation¶

The CDF answers the question "what is the probability that $X$ is at most $x$ ?" for every threshold $x$ at once. It collects probability from the left and describes the distribution of any real-valued random variable, whether its values form a list or a whole range.

Two CDFs: a staircase for the Binomial(3, 1/2) distribution with jumps of 1/8, 3/8, 3/8 and 1/8, and a smooth S-shaped curve for the standard normal distribution with its value at 1 marked.

The left panel is a staircase for $X\sim\operatorname{Bin}(3,\tfrac{1}{2})$ $Bin (3, \frac{1}{2})$ , the number of heads in three fair tosses. $F_X$ is flat between the possible values $0,1,2,3$ and jumps there by $\tfrac{1}{8},\tfrac{3}{8},\tfrac{3}{8},\tfrac{1}{8}$ , reaching the levels $\tfrac{1}{8},\tfrac{1}{2},\tfrac{7}{8},1$ ; each jump is the probability of that value, so $\mathbb{P}(X=1)=\tfrac{3}{8}$ $\frac{3}{8}$ . The right panel is the CDF of the standard normal distribution, which rises without any jump and has $F_X(1)\approx0.84$ $0.84$ .

Move $x$ : $F_X(x)$ collects all probability at or below $x$ . Masses give a staircase, a density a smooth curve, and the mixed example both a jump and a smooth rise.

Each property in the Definition has a probabilistic reason. $F_X$ is nondecreasing because $\{X\le x\}\subseteq\{X\le y\}$ for $x<y$ $y$ . The limits $0$ and $1$ hold because $\{X\le x\}$ shrinks to the impossible event as $x\to-\infty$ and grows to the whole sample space as $x\to\infty$ $\infty$ . Right-continuity holds because " $\le$ " includes the threshold itself, so at a jump the value is the top of the step: in the left panel $F_X(1)=\tfrac{1}{2}$ $\frac{1}{2}$ , not $\tfrac{1}{8}$ .

A common slip is to ignore the difference between $\le$ and $<$ . In the example $\mathbb{P}(X<1)=\tfrac{1}{8}$ while $\mathbb{P}(X\le1)=\tfrac{1}{2}$ $\frac{1}{2}$ ; for discrete variables the endpoints matter, hence the half-open interval $a<X\le b$ in the Key Results. For a discrete random variable, $F_X(x)$ is the sum of the PMF $p_X(t)$ over all values $t\le x$ $x$ .

Key Results¶

$\mathbb{P}(a<X\le b)=F_X(b)-F_X(a)$ $F_{X} (a)$ .
$\mathbb{P}(X=x)=F_X(x)-F_X(x-)$ $F_{X} (x -)$ , where $F_X(x-)$ is the left limit.

Examples¶

For $\operatorname{Bern}(0.3)$ , the CDF is $0$ for $x<0$ $0$ , $0.7$ for $0\le x<1$ $1$ , and $1$ for $x\ge1$ $1$ .
For $\operatorname{Unif}(0,1)$ , $F_X(0.4)=0.4$ and $\mathbb{P}(0.2<X\le0.4)=0.2$ $0.2$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3
Nicky van Foreest · Probability Distributions · Refresher

Aliases: CDF, distribution function, cumulative distribution, c.d.f. Symbols: element of, infinity, real numbers, subset

Last updated: 20 October 2026 03:02