# [SciPy-User] Kurtosis/Skewness

Robert Kern robert.kern@gmail....
Tue Mar 30 16:36:33 CDT 2010

```On Tue, Mar 30, 2010 at 16:32, nicky van foreest <vanforeest@gmail.com> wrote:
>> There are an infinite number of distributions that will have the same
>> skewness and kurtosis. However, it is reasonable to search for the
>> maximum entropy distribution satisfying those constraints. The normal
>> distribution is the maximum entropy distribution for a fixed mean and
>> variance.
>>
>> http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution
>>
>> The PDF will have the form:
>>
>>  pdf(x) = c * exp(- lagrange * (x ** arange(1, 5)))
>>
>> c is just the normalizing constant. You will have to find the lagrange
>> parameters that satisfy the mean, variance, skewness and kurtosis.
>> Sampling from this distribution will be tricky, though. You will have
>> to resort to general methods that are going to be pretty slow.
>
>
> This is of course a very good suggestion. However, mind that this
> claim is only true if the support of your desired distribution is the
> entire real axis. I recall that I once tried to find the maximum
> entropy distribution with given mean and variance, but such that the
> support was the positive reals (including 0), rather then the entire
> real line. This was less easy then I initially thought, and it is
> certainly not the normal distribution.

Right. Since he was starting with the normal distribution, I assume he
wanted something that went over the entire real axis.

--
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
-- Umberto Eco
```