Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 3 down vote accepted

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).

share|cite|improve this answer
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

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

share|cite|improve this answer
I think there may have been a few typos here. – whuber Sep 14 '10 at 16:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.