Lognormal Distribution

def. Lognormal Distribution. A random variable $X$ is a lognormal random variable iff

• $X=\ln (\text{N}(\mu ,{\sigma }^2))$ i.e. log of the Normal Distribution random variable
• PDF:

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

• $E(X)=e^{\mu +\frac{{\sigma }^2}{2}}$
• $Var(X)=(e^{{\sigma }^2}-1)e^{2\mu +{\sigma }^2}$
• CDF:

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

- Where $\Phi$ is the Standard [[Normal Distribution]] CDF.