Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Normal distribution ▷ Quantile function

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

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

Then, the quantile function of $X$ is

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

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

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

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

Because the cumulative distribution function (CDF) is strictly monotonically increasing, the quantile function is equal to the inverse of the CDF:

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

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

\[\label{eq:norm-qf-s2} \begin{split} p &= \frac{1}{2} \left[ 1 + \mathrm{erf}\left( \frac{x-\mu}{\sqrt{2} \sigma} \right) \right] \\ 2 p - 1 &= \mathrm{erf}\left( \frac{x-\mu}{\sqrt{2} \sigma} \right) \\ \mathrm{erf}^{-1}(2p-1) &= \frac{x-\mu}{\sqrt{2} \sigma} \\ x &= \sqrt{2}\sigma \cdot \mathrm{erf}^{-1}(2p-1) + \mu \; . \end{split}\]
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Metadata: ID: P87 | shortcut: norm-qf | author: JoramSoch | date: 2020-03-20, 04:47.