Cumulative distribution function

The cumulative distribution function (CDF) F(x) gives the probability that a random variable is at most x. Defined for every distribution, it converts probability questions into area calculations and underpins standard statistical tables.

ByHoang TruongUpdated July 3, 2026

What it is

See it move

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The cumulative distribution function F(x) = P(X ≤ x) rises monotonically from 0 to 1 as x increases. Any interval probability reduces to two evaluations: P(a < X ≤ b) = F(b) − F(a). Standard normal tables are simply a printed CDF for N(0,1), and quantiles such as the median are the x-values where F(x) reaches a target probability.

The formula

LaTeX

F(x) = P(X \le x) = \int_{-\infty}^{x} f(t)\, dt

Variables

$F (x)$: cumulative distribution function evaluated at x (dimensionless)
$f (t)$: probability density function (per unit of t)
$x$: value at which the CDF is evaluated (units of X)

For a discrete random variable, F(x) = sum of p(k) for all k ≤ x. F is non-decreasing, right-continuous, and ranges from 0 to 1.

LaTeX

P(a < X \le b) = F(b) - F(a)

Variables

$F (b)$: CDF evaluated at upper bound b (dimensionless)
$F (a)$: CDF evaluated at lower bound a (dimensionless)
$a, b$: lower and upper bounds of the interval (a < b) (units of X)

Any interval probability reduces to two CDF evaluations; standard normal tables apply this directly.

Check yourself

PracticeCORE

A continuous random variable X has cumulative distribution function F(x). Which of the following expressions correctly gives P(3 < X ≤ 7)?

Select an answer to check your understanding.