I was thinking about this last week at work when I was coding part of a model involving the parameters of a truncated lognormal distribution. The lognormal distribution definitely feels like it was named the wrong way round. What is a Lognormal Distribution?
We say that a Random Variable $X$ has a LogNormal Distribution that is:
$$ X \sim LogN( \mu , { \sigma }^2 ) $$ if: $$ Log (X) \sim N( \mu , { \sigma }^2 ) $$In other words, a Lognormal distribution is a distribution such that the log of the distribution is a normal distribution. It is not, as you might think, a distribution which is the log of the normal distribution. So if $Y \sim N( \mu , {\sigma}^2 ) $ then $Log ( Y ) $ is not a lognormal distribution, instead $ e ^ Y $ is a lognormal distribution. So to create a lognormal distribution, we don't take the log of the normal distribution, we take the exponential! Why does this matter?
Definitions are just definitions after all, and as long as everyone knows how something is defined and there is no ambiguity one definition is usually as good as another. In this case though, defining it in this way does have some ugly and unnatural consequences. For example, if we take the result that the sum of two independent normal distributions is also a normal distribution, i.e.
If: $$ X \sim N( {\mu}_1 , {{\sigma}_1}^2 ) , Y \sim N( {\mu}_2 , {{\sigma}_2}^2 ) $$ Then: $$ X + Y \sim N( {\mu}_1 + {\mu}_2 , {{\sigma}_1}^2 + {{\sigma}_2}^2 ) $$ Then applying this result to the lognormal distribution, we get: If $ X \sim LogN( {\mu}_1 , {{\sigma}_1}^2 ) $ and $ Y \sim LogN( {\mu}_2 , {{\sigma}_2}^2 ) $ assuming independence, Then:$$ XY \sim LogN( {\mu}_1 + {\mu}_2 , {{\sigma}_1}^2 + {{\sigma}_2}^2 ) $$ Maybe this doesn't look too bad to you. But what if I replace $X$ and $Y$ with ${LogN}_1$ and ${LogN}_2$? Then we get: $$ {LogN}_1 * {LogN}_2 \sim LogN( {\mu}_1 + {\mu}_2 , {{\sigma}_1}^2 + {{\sigma}_2}^2 ) $$ This should definitely look wrong to you! Remember that for a standard logarithm: $$ Log (AB) = Log(A) + Log(B) $$ Instead we have an identity that looks much more like an exponential:$$ e^A * e^B = e^{ (A + B ) } $$ And that's precisely because we are dealing with an exponential! The lognormal distribution is simply the exponential of the normal, which is a much more natural way of phrasing it than to say that the lognormal distribution is a distribution such that the logarithm of the distribution is a normal distribution. So we have two reasons why the Lognormal Distbribution should have been called the Exponential Normal Distribution (Or possibly the XNormal Distribution for short). The identity above makes perfect sense when using exponentials, and we would have a naming convention that is much more natural.
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AuthorI work as a pricing actuary at a reinsurer in London. Categories
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January 2020
