#### Subject 10. The Lognormal Distribution

The key properties of the normal distribution have been presented in the LOS above. As a summary, some of the key properties of this distribution are:

• It is symmetrical about the mean.
• It has zero skewness.
• It has a kurtosis of 3.

A random variable, Y, follows a lognormal distribution if its natural logarithm, lnY, is normally distributed. You can think of the term lognormal as "the log is normal." For example, suppose X is a normal random variable, and Y = eX. Therefore, LnY = Ln(eX) = X. Because X is normally distributed, Y follows a lognormal distribution.

• Like the normal distribution, the lognormal distribution is completely described by two parameters: mean and variance.
• Unlike the normal distribution, the lognormal distribution is defined in terms of the parameters of the associated normal distribution. Note that the mean of Y is not equal to the mean of X, and the variance of Y is not equal to the variance of X. In contrast, the normal distribution is defined by its own mean and variance.
• The lognormal distribution is bounded below by 0. In contrast, the normal distribution extends to negative infinity without limit.
• The lognormal distribution is skewed to the right (i.e., it has a long right tail). In contrast, the normal distribution is bell-shaped (i.e., it is symmetrical). The reverse is also true; if a random variable Y follows a lognormal distribution, then its natural logarithm, lnY, is normally distributed.