# The pdf of the ratio of two lognormal distributions [closed]

What is the pdf of the ratio of two independent lognormal distributions? Why is $$log(X)$$ normal when $$X$$ is lognormal?

• Hi, welcome to CV. If this is homework then you should add the 'self-study' tag. It's always good to provide some context for your questions and to make sure you've searched the site for existing answers that might address yours (particularly the second question). Regarding the first question, it might be worth seeing if you can find the distribution of $\log(X/Y)$.
– epp
Jun 16 '19 at 7:10

Let $$X,Y$$ be independent log-normal random variables. Then by definition, $$\log X$$ and $$\log Y$$ are normally distributed.
Let $$Z= \frac{X}{Y}$$ be their ratio. Then clearly we have
$$\log Z = \log \left( \frac{X}{Y} \right) = \log X - \log Y$$
It follows that $$\log Z$$ is normally distributed, being the difference of two independent normally distributed random variables, and hence $$Z = \frac{X}{Y}$$ is log-normal.