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 Log-normal Distribution?**

$$ X \sim LogN( \mu , { \sigma }^2 ) $$

if: $$ Log (X) \sim N( \mu , { \sigma }^2 ) $$In other words, a Log-normal 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?**

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 X-Normal Distribution for short). The identity above makes perfect sense when using exponentials, and we would have a naming convention that is much more natural.