Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Log-normal distribution ▷ Probability density 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 probability density function of $X$ is given by:

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

Proof: A log-normally distributed random variable is defined as the exponential function of a normal random variable:

\[\label{eq:lognorm-def} Y \sim \mathcal{N}(\mu,\sigma^2) \; \quad \Rightarrow \quad X = \mathrm{exp} (Y) \sim \ln \mathcal{N}(\mu, \sigma^2) \; .\]

The probability density function of the normal distribution is

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

Writing $X$ as a function of $Y$, we have

\[\label{eq:X-Y} X = g(Y) = \mathrm{exp} (Y) \;\]

with the inverse function

\[\label{eq:Y-X} Y = g^{-1}(X) = \ln (X) \; .\]

Because the derivative of $\exp (Y)$ is always positive, $g(Y)$ is strictly increasing and we can calculate the probability density function of a strictly increasing function as

\[\label{eq:pdf-sifct} f_X(x) = \left\{ \begin{array}{rl} f_Y(g^{-1}(x)) \, \frac{\mathrm{d}g^{-1}(x)}{\mathrm{d}x} \; , & \text{if} \; x \in \mathcal{X} \\ 0 \; , & \text{if} \; x \notin \mathcal{X} \end{array} \right.\]

where $\mathcal{X} = \left\lbrace x = g(y): y \in \mathcal{Y} \right\rbrace$. With the probability density function of the normal distribution, we have

\[\label{eq:pdf-X} \begin{split} f_X(x) &= f_Y(g^{-1}(x))\cdot \frac{dg^{-1}(x)}{dx} \\ &= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{g^{-1}(x)-\mu}{\sigma} \right)^2 \right] \cdot \frac{\mathrm{d}g^{-1}(x)}{\mathrm{d}x} \\ &= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{(\ln x)-\mu}{\sigma} \right)^2 \right] \cdot \frac{\mathrm{d}(\ln x)}{\mathrm{d}x} \\ &= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{ \ln x -\mu}{\sigma} \right)^2 \right] \cdot \frac{1}{x} \\ &= \frac{1}{x \sigma \sqrt{2 \pi}} \cdot \exp \left[ - \frac{\left( \ln x -\mu\right)^2}{2 \sigma^2} \right] \end{split}\]

which is the probability density function of the log-normal distribution.

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Metadata: ID: P310 | shortcut: lognorm-pdf | author: majapavlo | date: 2022-02-13, 10:05.