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Distribution 11.5 The Extreme Value (aka Gumbel) Distribution

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11.4 The log Normal Distribution

Parameters: $\boldsymbol{\theta}=(\xi,\sigma^{2})$ with $\xi\in\mathbb{R}$ and $\sigma^{2}>0$ .

$f_{X}(x;\boldsymbol{\theta})=\frac{1}{\sqrt{2\pi}}\frac{1}{x\sigma}\exp\left\{% -\frac{(\log x-\xi)^{2}}{2\sigma^{2}}\right\}$ for $0<x<\infty$ ,
$\operatorname{\mathsf{E}}\left[{X}\right]=\exp\left(\frac{\sigma^{2}}{2}+\xi\right)$ ,
${\operatorname{\mathsf{Var}}}\left[{X}\right]=\left\{\exp(\sigma^{2})-1\right% \}\exp(\sigma^{2}+2\xi)$ .

We write $X\sim\log\operatorname{\mathsf{N}}(\xi,\sigma^{2})$ .

Transformations: If $Y\sim\operatorname{\mathsf{N}}(\xi,\sigma^{2})$ , then $X=\exp(Y)$ has a log normal distribution with parameters $(\xi,\sigma^{2})$ .

Quiz: To prove this would you use the cdf method or the density method? Much easier using the density method.

Usage: Positive quantities such as the concentration of a pollutant or time to an event.