# Characteristics of a Gaussian on a logarithmic scale?

I have data modelled to be Gaussian-distributed, and then displayed on a logarithmic scale and like to know some properties of the displayed data.

Effektively, I ignore that a Gaussian $X$ (with positive mean) may be negative and need mean, standard deviation and full width at half maximum of $\log X$. Unfortunately, the probability density is sufficiently ugly to prevent me from calculating this myself. Has this distribution a special name that I might look for?

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No, it does not have a standard name. By analogy to the lognormal distribution it might be dubbed the "exponential-normal" distribution. But the analogy is imperfect, because you have to truncate the normal part at zero.

The integrals to compute the moments (or, equivalently, the characteristic function) do not have closed forms. You will need to compute them numerically. They are obtained by integrating

$\frac{1}{\sqrt{2 \pi } \sigma } x^k e^{x-\frac{\left(e^x-\mu \right)^2}{2 \sigma ^2}} dx$

over the entire real line. (This comes from the pdf for a Normal distribution of $y$ with mean $\mu$ and standard deviation $\sigma$ and substituting $y = exp(x)$.) $k = 1$ yields the mean; $k = 2$ gives the second moment; subtracting the square of the mean from that is the variance; and the square root of the variance is the standard deviation. Finding the width at half maximum also requires numerical methods (but convergence should be rapid).

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Thanks, I feared that this might be the case. =) On to the numerics! – Jens Sep 14 '10 at 18:57

You can work out the distribution using the transformation theorem. If you have a normal density truncated to positive values, the log of that distribution has PDF

$\frac{2}{\sqrt{2\pi}} \exp(x -\exp(2x)/2)$

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I think there may have been a few typos here. – whuber Sep 14 '10 at 16:29