Bruce Sherman

2022-09-09

Find distribution mean from the mean and sd of the log
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?

Conor Daniel

You are asking, if I know ${\int }_{\mathbb{R}}\mathrm{ln}\left(x\right)f\left(x\right)dx$ and ${\int }_{\mathbb{R}}{\mathrm{ln}}^{2}\left(x\right)f\left(x\right)dx$, how can I find out what ${\int }_{\mathbb{R}}xf\left(x\right)dx$ 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 $\mathrm{ln}\left(X\right)$ is log-normal, with the transformation straight forward:
$\begin{array}{rl}m& ={e}^{\mu +{\sigma }^{2}/2}\\ v& =\left({e}^{{\sigma }^{2}}-1\right){e}^{2\mu +{\sigma }^{2}},\end{array}$
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.

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