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?

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

Beginner2022-09-10Added 11 answers

You are asking, if I know ${\int}_{\mathbb{R}}\mathrm{ln}(x)f(x)dx$ and ${\int}_{\mathbb{R}}{\mathrm{ln}}^{2}(x)f(x)dx$, how can I find out what ${\int}_{\mathbb{R}}xf(x)dx$ 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 $\mathrm{ln}(X)$ is log-normal, with the transformation straight forward:

$$\begin{array}{rl}m& ={e}^{\mu +{\sigma}^{2}/2}\\ v& =({e}^{{\sigma}^{2}}-1){e}^{2\mu +{\sigma}^{2}},\end{array}$$

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.

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 $\mathrm{ln}(X)$ is log-normal, with the transformation straight forward:

$$\begin{array}{rl}m& ={e}^{\mu +{\sigma}^{2}/2}\\ v& =({e}^{{\sigma}^{2}}-1){e}^{2\mu +{\sigma}^{2}},\end{array}$$

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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