Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Cumulative distribution function ▷ Cumulative distribution function of continuous random variable

Theorem: Let $X$ be a continuous random variable with possible values $\mathcal{X}$ and probability density function $f_X(x)$. Then, the cumulative distribution function of $X$ is

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

Proof: The cumulative distribution function of a random variable $X$ is defined as the probability that $X$ is smaller than $x$:

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

The probability density function of a continuous random variable $X$ can be used to calculate the probability that $X$ falls into a particular interval $A$:

\[\label{eq:pdf} \mathrm{Pr}(X \in A) = \int_{A} f_X(x) \, \mathrm{d}x \; .\]

Taking these two definitions together, we have:

\[\label{eq:cdf-pdf-qed} \begin{split} F_X(x) &\overset{\eqref{eq:cdf}}{=} \mathrm{Pr}(X \in (-\infty, x]) \\ &\overset{\eqref{eq:pdf}}{=} \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; . \end{split}\]
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Metadata: ID: P190 | shortcut: cdf-pdf | author: JoramSoch | date: 2020-11-12, 06:33.