## 04 March 2009

### Why isn't it expnormal?

We say that a random variable X has a lognormal distribution if its logarithm, Y = log X, is normally distributed. The normal distribution often occurs when a random variable comes about by combining a bunch of small independent contributions, but those contributions combine additively; when the combination is multiplicative instead, lognormals occur. For example, lognormal distributions often occur in models of financial markets.

But of course X = exp Y, so the variable we care about is the exponential of a normal. Why isn't it called expnormal?

Mike said...

I don't know if you've seen this paper, but they make the case that it's actually the lognormal distribution that's ubiquitous in primary experimental data. In biology at least, I think it makes sense, since most features are generated by growth processes where errors tend to have multiplicative rather than additive effects.

Michael Lugo said...

Mike,

I hadn't seen that, and lack the time to read it right now, but thanks for the pointer. The central point seems reasonable at first glance; when the variance is small the lognormal approaches the normal, so it's easy to get them confused. Anonymous said...

I agree that it seems backward. You might reasonably think that lognormal means "log of a normal." But instead it means "distribution whose log is normal."

Although this is confusing, it is conventional in probability. A [foo][bar] distribution is one such that when you apply function [foo], you get distribution [bar]. For example, logit-normal is sometimes used for a distribution such that its logit is normal, i.e. the inverse logit of a normal distribution.

Barry said...

I'm guessing lognormal means that it looks like a normal distribution on log graph paper.

Anybody else remember log graph paper or am I dating myself. Anonymous said...

It's the same pattern as log-convex. In general I feel like saying X is noun-adjective should mean that the noun of X is adjective, but I'm having trouble coming up with good real-world examples.

Does that pattern sound right to you when the noun is not the name of a function like log? For example, if someone were to coin the word "derivative-increasing", would you interpret it as nonnegative or convex? I'd definitely choose convex, but maybe it's just me.