Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Log-normal distribution ▷ Median

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

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

Then, the median of $X$ is

\[\label{eq:lognorm-med} \mathrm{median}(X) = e^\mu \; .\]

Proof: The median is the value at which the cumulative distribution function is $1/2$:

\[\label{eq:median} F_X(\mathrm{median}(X)) = \frac{1}{2} \; .\]

The cumulative distribution function of the lognormal distribution is

\[\label{eq:lognorm-cdf} F_X(x) = \frac{1}{2} \left[ 1 + \mathrm{erf} \left( \frac{\ln (x)-\mu}{\sigma \sqrt{2}} \right) \right]\]

where $\mathrm{erf}(x)$ is the error function defined as

\[\label{eq:erf} \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp(-t^2) \, \mathrm{d}t \; .\]

Thus, the inverse CDF is

\[\label{eq:lognorm-cdf-inv} \begin{split} \ln(x) &= \sigma \sqrt{2} \cdot \mathrm{erf}^{-1}(2p-1) + \mu \\ x &= \mathrm{exp} \left[ \sigma \sqrt{2} \cdot \mathrm{erf}^{-1}(2p-1) + \mu \right] \end{split}\]

where $\mathrm{erf}^{-1}(x)$ is the inverse error function. Setting $p = 1/2$, we obtain:

\[\label{eq:lognorm-med-qed} \begin{split} \ln \left[ \mathrm{median}(X) \right] &= \sigma \sqrt{2} \cdot \mathrm{erf}^{-1}(0) + \mu \\ \mathrm{median}(X) &= e^\mu \; . \end{split}\]
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Metadata: ID: P306 | shortcut: lognorm-med | author: majapavlo | date: 2022-02-07, 22:33.