4.5.2 The Lognormal Distribution

lognormal distribution

A nonnegative rv $X$ is said to have a lognormal distribution if the rv $Y=\ln \left(X\right)$ has a normal distribution. The resulting pdf of a lognormal rv when $\ln \left(X\right)$ is normally distributed with parameters $\mu$ and $\sigma$ is

$$f\left(x;\mu ,\sigma \right)=\{\begin{array}{cc}\frac{1}{\sqrt{2\pi }\sigma x}{e}^{-{\left[\ln \left(x\right)-\mu \right]}^2/\left(2{\sigma }^2\right)}& x\ge 0\\ 0& x<0\end{array}$$

Be careful here; the parameters $\mu$ and $\sigma$ are not the mean and standard deviation of $X$ but of $\ln \left(X\right)$. The mean and variance of $X$ can be shown to be

$$E\left(X\right)=e^{\mu +{\sigma }^2/2}V\left(X\right)=e^{2\mu +{\sigma }^2}\cdot \left(e^{{\sigma }^2}-1\right)$$

In Chapter 5, we will present a theoretical justification for this distribution in connection with the Central Limit Theorem. But as with other distributions, the lognormal can be used as a model even in the absence of such justification. Figure 4.30 illustrates graphs of the lognormal pdf; although a normal curve is symmetric, a lognormal curve has a positive skew.

Because $\ln \left(X\right)$ has a normal distribution, the cdf of $X$ can be expressed in terms of the cdf $\Phi \left(z\right)$ of a standard normal rv $Z$.

Proposition

$$\begin{array}{rl}& F(x;\mu ,\sigma )\\ & =P(X\le x)=P\left[\ln (X)\le \ln (x)\right]\\ & =P\left(Z\le \frac{\ln (x)-\mu }{\sigma }\right)\\ & =\Phi \left(\frac{\ln (x)-\mu }{\sigma }\right),x\ge 0.\end{array}\text{\hspace{1em}(4.13)}$$

EX 4.27