Scaling a Lognormal Distribution

Question

I am wondering how I can apply a scale factor to a lognormal distribution with a mean of 1.

Starting with the lognormal distribution $X$~$Lognormal(\mu=-0.5, \sigma=1)$.

Then the mean of the distribution is $e^{\mu+\frac{\sigma^2}{2}}=e^{-0.5+\frac{1^2}{2}}=1$.

This distribution is a normalized version of population distribution. If we know that the population mean is actually 100. How can we scale the lognormal distribution so that it has the same shape, but a mean of 100?

Answer 1

By inspection, $\mu$ is a scale parameter in the lognormal, $\sigma$ is a shape parameter.

Clearly, then, if you want to maintain the same shape and change the scale, you change $\mu$ to $\mu'$ say, such that $e^{\mu'+\frac12\sigma^2}=100$ with the same $\sigma$ as before.