R Notebook

In lognorm: Functions for the Lognormal Distribution

Lognormal parameters for mode and upper quantile

Mode: $mode = e^{\mu - \sigma^2}$

Upper quantile: $upper = e^{\mu + f \sigma}$.

log transformed: $$ m = \mu - \sigma^2 \ u = \mu + f \sigma $$ Can solve for $\sigma$:

$$ u - m = f \sigma + \sigma^2 \ 0 = \sigma^2 + f \sigma - (u-m) $$ Solution for quadratic equations of the form $x^2 + px + q$ are $x_{1,1} = -p/2 \pm \sqrt{p^2/4 - q}$.

$$ \sigma_{1,2} = -f/2 \pm \sqrt{f^2/4 + (u-m)} $$

Since $u$ is an upper quantile, $(u-m)>0$ and the root is larger than $f/2$. Hence, there is one positive solution:

$$ \sigma = -f/2 + \sqrt{f^2/4 + (u-m)} \ \mu = m + \sigma^2 $$

Lognormal parameters for Mean and upper quantile

Mode: $mean = e^{\mu + \sigma^2/2}$

Upper quantile: $upper = e^{\mu + f \sigma}$.

log transformed: $$ m = \mu + \sigma^2/2 \ u = \mu + f \sigma $$ Can solve for $\sigma$:

$$ u - m = f \sigma - \sigma^2/2 \ 0 = \sigma^2 - 2 f \sigma + 2(u-m) \ \sigma_{1,2} = f \pm \sqrt{f^2 - 2(u-m)} $$

Hence, there are two positive solutions. We are interested in the one that has the smaller standard deviation.

$$ \sigma = f - \sqrt{f^2 - 2(u-m)} \ \mu = m - \sigma^2/2 $$

lognorm documentation built on Nov. 22, 2021, 1:07 a.m.