Normal and Log-Normal Probability Density Functions with Identical Mean and Standard Deviation

This Demonstration compares a log-normal distribution probability density function (PDF) with parameters and with a normal distribution PDF with parameters and . Both PDFs have identical mean and standard deviation. Since the log-normal distribution is defined for positive values, it is not possible to have an equivalent normal distribution with negative mean.

The PDF of a log-normal distribution variable with parameters and is given by

,

where denotes the natural logarithm, and and are related to the variable mean and its standard deviation σ according to

,

.

The PDF for a normal distribution random variable with parameters and is given by

.

,

and similarly, the standard deviation squared, of , is

.

As can be verified using the built-in Mathematica function Expectation, the mean and standard deviation for both the log-normal distribution and normal distribution are and .