**Quantitative Methods: Application**

**Reading 9. Common Probability Distributions**

**Learning Outcome Statements**

o. explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices;

*CFA Curriculum, 2020, Volume 1*

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### Subject 10. The Lognormal Distribution

- 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 = e

^{X}. Therefore, LnY = Ln(e

^{X}) = 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.

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**User Contributed Comments**
6

You need to log in first to add your comment. ###### rufi

this is a good tool for BSOPM

###### bahodir

what is BSOPM?

###### bobert

Black-Scholes Option Pricing Model

###### Seemorr

What kind of variable would be lognormally distributed, but not normally?

###### riouxcf

Some variables which have frequent outliers can be made more normal by taking the log. The normal distribution tends to underestimate extremes.

###### czar

Seemor: stock prices (log) and stock returns (normal) as stock prices lowest value can only be 0 while returns can be negative