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

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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.

Where it fits

TopicProbability & DistributionsAdvancedSubjectData Analysis & StatisticsAdvanced

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.