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Lognormal distribution

A positive, right-skewed quantity whose logarithm is normal - the default model for claim severity and asset prices.

Parameters and support

$\mu$

mean of the UNDERLYING normal - $\mu \in \mathbb{R}$

$\sigma$

standard deviation of the UNDERLYING normal - $\sigma >0$

Support

$x>0$

The formulas

f(x)

$$f(x)=\frac{1}{x\sigma \sqrt{2\pi }}\exp \left(-\frac{(\ln x-\mu {)}^2}{2{\sigma }^2}\right)$$

F(x)

$$F(x)=\Phi \left(\frac{\ln x-\mu }{\sigma }\right)$$

Mean

$$E[X]=e^{\mu +{\sigma }^2/2}$$

Variance

$$\mathrm{Var}(X)=\left(e^{{\sigma }^2}-1\right)e^{2\mu +{\sigma }^2}$$

MGF

Not examinable for this distribution.

The MGF does not exist for any t > 0 - the tail is too heavy. Use the moment formula E[Xᵏ] = exp(kμ + k²σ²/2) instead.

Where the moments come from

1. $X=e^Y$ with $Y\sim N(\mu ,{\sigma }^2)$, so $P(X\le x)=P(Y\le \ln x)=\Phi \left(\frac{\ln x-\mu }{\sigma }\right)$.
2. $E[X^k]=E[e^{kY}]=M_Y(k)=e^{k\mu +k^2{\sigma }^2/2}$ - the normal MGF evaluated at $k$.
3. $k=1$ gives $E[X]=e^{\mu +{\sigma }^2/2}$; $k=2$ gives $E[X^2]=e^{2\mu +2{\sigma }^2}$.
4. $\mathrm{Var}(X)=e^{2\mu +2{\sigma }^2}-e^{2\mu +{\sigma }^2}=(e^{{\sigma }^2}-1)e^{2\mu +{\sigma }^2}$.
5. The median is $e^{\mu }$ because $\Phi (0)=0.5$ at $\ln x=\mu$.

Worked example

The mean, variance, CDF and moment generating function above are re-derived numerically from this distribution’s own density on every test run - summed over the support for a discrete distribution, integrated by quadrature for a continuous one - and compared with the closed forms printed here. A typo on this page fails the build.

Traps

• Reporting e^μ as the mean - that is the MEDIAN.
• Using σ² where σ belongs inside the Φ argument.
• Looking for an MGF; it does not exist. Use the k-th moment formula.

Related

• Normal distribution
• Gamma distribution
• Pareto distribution

Drill this: the Exam P question bank has original questions on this distribution, and the recall trainer builds its prompts from exactly the formulas above.