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Cumulative Distribution Function (CDF) ○꠹｜Definition｜1st｜20260604203856-00-⌔

Cumulative distribution function - Wikipedia

Cumulative distribution function

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In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable $X$, or just distribution function of $X$, evaluated at $x$, is the probability that $X$ will take a value less than or equal to $x$.¹

Every probability distribution supported on the real numbers, discrete or “mixed” as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) $F:\mathbb{R}\to [0,1]$ satisfying $\underset{x\to -\infty }{\lim }F(x)=0$ and $\underset{x\to \infty }{\lim }F(x)=1$.

In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to $x$. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

Printed 2026-06-28.

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Footnotes

1. Deisenroth, Marc Peter; Faisal, A. Aldo; Ong, Cheng Soon (2020). Mathematics for Machine Learning. Cambridge University Press. p. 181. ISBN 9781108455145. ↩

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