Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability functions ▷ Probability density function in terms of cumulative distribution function

Theorem: Let $X$ be a continuous random variable. Then, the probability distribution function of $X$ is the first derivative of the cumulative distribution function of $X$:

\[\label{eq:pdf-cdf} f_X(x) = \frac{\mathrm{d}F_X(x)}{\mathrm{d}x} \; .\]

Proof: The cumulative distribution function in terms of the probability density function of a continuous random variable is given by:

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

Taking the derivative with respect to $x$, we have:

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

The fundamental theorem of calculus states that, if $f(x)$ is a continuous real-valued function defined on the interval $[a,b]$, then it holds that

\[\label{eq:FToC-1st} F(x) = \int_{a}^{x} f(t) \, \mathrm{d}t \quad \Rightarrow \quad F'(x) = f(x) \quad \text{for all} \quad x \in (a,b) \; .\]

Applying \eqref{eq:FToC-1st} to \eqref{eq:cdf-pdf}, it follows that

\[\label{eq:pdf-cdf-qed} F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \quad \Rightarrow \quad \frac{\mathrm{d}F_X(x)}{\mathrm{d}x} = f_X(x) \quad \text{for all} \quad x \in \mathbb{R} \; .\]
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Metadata: ID: P191 | shortcut: pdf-cdf | author: JoramSoch | date: 2020-11-12, 07:19.