Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Log-normal distribution ▷ Quantile function

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 quantile function of $X$ is

\[\label{eq:lognorm-qf} Q_X(p) = \exp( \mu + \sqrt{2} \sigma \cdot \mathrm{erf}^{-1}(2p-1) )\]

where $\mathrm{erf}^{-1}(x)$ is the inverse error function.

Proof: The cumulative distribution function of the log-normal distribution is:

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

From this point forward, the proof is similar to the derivation of the quantile function for the normal distribution. Because the cumulative distribution function (CDF) is strictly monotonically increasing, the quantile function is equal to the inverse of the CDF:

\[\label{eq:lognorm-qf-s1} Q_X(p) = F_X^{-1}(x) \; .\]

This can be derived by rearranging equation \eqref{eq:lognorm-cdf}:

\[\label{eq:lognorm-qf-s2} \begin{split} p &= \frac{1}{2} \left[ 1 + \mathrm{erf}\left( \frac{\ln x-\mu}{\sqrt{2} \sigma} \right) \right] \\ 2 p - 1 &= \mathrm{erf}\left( \frac{\ln x-\mu}{\sqrt{2} \sigma} \right) \\ \mathrm{erf}^{-1}(2p-1) &= \frac{\ln x-\mu}{\sqrt{2} \sigma} \\ x &= \exp(\mu + \sqrt{2}\sigma \cdot \mathrm{erf}^{-1}(2p-1) ) \; . \end{split}\]

Metadata: ID: P326 | shortcut: lognorm-qf | author: majapavlo | date: 2022-07-09, 11:05.