How to reparametize a distribution?

Question

I have a hard time understanding what's the purpose of reparametization and how to do it. My prof briefly talked about it without giving us an example and asked us to reparametize the Log Normal distribution but I have no idea how to start it.

Thank you in advance for your help!

Answer 1

A log-normal distribution is usually parametrised by the mean $\mu$ and variance $\sigma^2$ of the underlying normal distribution.

But you could do something else, for example by the actual mean $m$ and variance $v$ of the log-normal. If you try this, you need to find $\mu$ and $\sigma^2$ in terms of $m$ and $v$, and then substitute that into the pdf or CDF. You might start with

• $m=\exp\left(\mu+\frac{\sigma^2}{2}\right)$
• $v=\left(\exp(\sigma^2)-1\right) \exp\left(2\mu+\sigma^2\right)$
• so $\exp(\sigma^2)=1+\frac{v}{\mu^2}$
• and so on