Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Normal distribution ▷ Full width at half maximum

Theorem: Let $X$ be a random variable following a normal distribution:

\[\label{eq:norm} X \sim \mathcal{N}(\mu, \sigma^2) \; .\]

Then, the full width at half maximum (FWHM) of $X$ is

\[\label{eq:norm-fwhm} \mathrm{FWHM}(X) = 2 \sqrt{2 \ln 2} \sigma \; .\]

Proof: The probability density function of the normal distribution is

\[\label{eq:norm-pdf} f_X(x) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right]\]

and the mode of the normal distribution is

\[\label{eq:norm-mode} \mathrm{mode}(X) = \mu \; ,\]

such that

\[\label{eq:norm-pdf-max} f_\mathrm{max} = f_X(\mathrm{mode}(X)) \overset{\eqref{eq:norm-mode}}{=} f_X(\mu) \overset{\eqref{eq:norm-pdf}}{=} \frac{1}{\sqrt{2 \pi} \sigma} \; .\]

The FWHM bounds satisfy the equation

\[\label{eq:x-FHWM} f_X(x_\mathrm{FWHM}) = \frac{1}{2} f_\mathrm{max} \overset{\eqref{eq:norm-pdf-max}}{=} \frac{1}{2 \sqrt{2 \pi} \sigma} \; .\]

Using \eqref{eq:norm-pdf}, we can develop this equation as follows:

\[\label{eq:x-FHWM-s1} \begin{split} \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x_\mathrm{FWHM}-\mu}{\sigma} \right)^2 \right] &= \frac{1}{2 \sqrt{2 \pi} \sigma} \\ \exp \left[ -\frac{1}{2} \left( \frac{x_\mathrm{FWHM}-\mu}{\sigma} \right)^2 \right] &= \frac{1}{2} \\ -\frac{1}{2} \left( \frac{x_\mathrm{FWHM}-\mu}{\sigma} \right)^2 &= \ln \frac{1}{2} \\ \left( \frac{x_\mathrm{FWHM}-\mu}{\sigma} \right)^2 &= -2 \ln \frac{1}{2} \\ \frac{x_\mathrm{FWHM}-\mu}{\sigma} &= \pm \sqrt{2 \ln 2} \\ x_\mathrm{FWHM}-\mu &= \pm \sqrt{2 \ln 2} \sigma \\ x_\mathrm{FWHM} &= \pm \sqrt{2 \ln 2} \sigma + \mu \; . \end{split}\]

This implies the following two solutions for $x_\mathrm{FWHM}$

\[\label{eq:x-FHWM-s2} \begin{split} x_1 &= \mu - \sqrt{2 \ln 2} \sigma \\ x_2 &= \mu + \sqrt{2 \ln 2} \sigma \; , \end{split}\]

such that the full width at half maximum of $X$ is

\[\label{eq:norm-fwhm-qed} \begin{split} \mathrm{FWHM}(X) &= \Delta x = x_2 - x_1 \\ &\overset{\eqref{eq:x-FHWM-s2}}{=} \left( \mu + \sqrt{2 \ln 2} \sigma \right) - \left( \mu - \sqrt{2 \ln 2} \sigma \right) \\ &= 2 \sqrt{2 \ln 2} \sigma \; . \end{split}\]
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Metadata: ID: P152 | shortcut: norm-fwhm | author: JoramSoch | date: 2020-08-19, 06:39.