Find distribution mean from the mean and sd of the log

Question

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log.

Given the mean and standard deviation of the log, how do I find the mean of the actual distribution?

Answer 1

You are asking, if I know $\int_\mathbb{R} \ln(x) f(x) dx$ and $\int_\mathbb{R} \ln^2(x) f(x) dx$, how can I find out what $\int_\mathbb{R} x f(x) dx$ is.

I don't think there is a general answer. Let $m, v$ denote mean and variance of the log-distribution. For example, if $X$ is normal, then $\ln(X)$ is log-normal, with the transformation straight forward: $$ \begin{split} m &= e^{\mu + \sigma^2/2} \\ v &= \left(e^{\sigma^2}-1\right) e^{2\mu + \sigma^2}, \end{split} $$ which needs to be solved for $\mu, \sigma$.

But if $X$ is uniform, say, the transformation would be different. You need to know something about the distribution of $X$ to make a reasonable estimate.