Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability functions ▷ Cumulative distribution function

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

\[\label{eq:cdf-cont} F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; .\]
 
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Metadata: ID: D13 | shortcut: cdf | author: JoramSoch | date: 2020-02-17, 22:07.