# Log Normal Distribution

A probability distribution \(LN(\mu, \sigma)\) having the PDF

The parameters \(\mu\) and \(\sigma\) are equal to the mean and standard deviation of the logarithm of \(x\) respectively.

If \(\mu\) is equal to zero and \(\sigma\) is equal to one then it is known as the

\[
p_{\mu,\sigma}(x) = \frac{\mathrm{d}}{\mathrm{d}x} \Phi_{\mu,\sigma}(\ln x) = \frac{1}{\sqrt{2\pi}\sigma x} e^{-\tfrac{(\ln x - \mu)^2}{2\sigma^2}}
\]

that governs the probability of products of random variables, where \(\Phi\) is the CDF of the normal distribution.The parameters \(\mu\) and \(\sigma\) are equal to the mean and standard deviation of the logarithm of \(x\) respectively.

If \(\mu\) is equal to zero and \(\sigma\) is equal to one then it is known as the

*standard*log normal distribution.