Definition: Cumulative distribution function

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Cumulative distribution function ▷ Definition

Definition: The cumulative distribution function (CDF) of a random variable $X$ at a given value $x$ is defined as the probability that $X$ is smaller than $x$:

\[\label{eq:cdf} F_X(x) = \mathrm{Pr}(X \leq x) \; .\]

1) If $X$ is a discrete random variable with possible outcomes $\mathcal{X}$ and the probability mass function $f_X(x)$, then the cumulative distribution function is the function $F_X(x): \mathbb{R} \to [0,1]$ with

\[\label{eq:cdf-disc} F_X(x) = \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; .\]

2) If $X$ is a continuous random variable with possible outcomes $\mathcal{X}$ and the probability density function $f_X(x)$, then the cumulative distribution function is the function $F_X(x): \mathbb{R} \to [0,1]$ with

\[\label{eq:cdf-cont} F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; .\]

 

Sources:

• Wikipedia (2020): "Cumulative distribution function"; in: Wikipedia, the free encyclopedia, retrieved on 2020-02-17; URL: https://en.wikipedia.org/wiki/Cumulative_distribution_function#Definition.

Metadata: ID: D13 | shortcut: cdf | author: JoramSoch | date: 2020-02-17, 22:07.