| Parameters |
$$µ \in \mathbb{R} $$ $$\sigma>0$$ |
| Support | $$x \in (0,+\infty) \quad and \quad p \in \:]0,1 [$$ |
| $$ PDF = \frac{1} {x \sigma \sqrt{2 \pi}} e^{ -\frac {(ln(x)-µ)}{ 2 \sigma^2}} $$ | |
| CDF |
$$ CDF = \frac{1}{2} + \frac{1}{2} er f \left[ \frac{ln(x)-µ}{\sqrt{2} \sigma} \right] $$ $$ with \qquad er \, f(.) \, , \qquad the \, error \, function $$ |
| CDF-1 |
$$ CDF^{-1} = \sqrt{ln \left( \frac{\sigma ^2}{µ^2} \right) + 1} \times \left[ \sqrt2 er f^{-1} (2 \times p-1 \right] + ln \left( \frac {µ^2} {\sigma^2+µ^2} \right) $$ $$ with \qquad er f^{-1}(.) , \qquad the \, inverse \, error \, function $$ |
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