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?

Bruce Sherman

Bruce Sherman

Answered question

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?

Answer & Explanation

Conor Daniel

Conor Daniel

Beginner2022-09-10Added 11 answers

You are asking, if I know R ln ( x ) f ( x ) d x and R ln 2 ( x ) f ( x ) d x, how can I find out what R x f ( x ) d x 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:
m = e μ + σ 2 / 2 v = ( e σ 2 1 ) e 2 μ + σ 2 ,
which needs to be solved for μ , σ
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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