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

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closed as off-topic by kjetil b halvorsen, Michael Chernick, Peter Flom Jun 17 at 8:29

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  • "Self-study questions (including textbook exercises, old exam papers, and homework) that seek to understand the concepts are welcome, but those that demand a solution need to indicate clearly at what step help or advice are needed. For help writing a good self-study question, please visit the meta pages." – kjetil b halvorsen, Michael Chernick, Peter Flom
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    $\begingroup$ 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 $\log(X/Y)$. $\endgroup$ – StatsPlease Jun 16 at 7:10
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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.

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