Re: [Phys-L] tutorial on exponential mixtures

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding:

> Reminds multiple nuclide decay.
>
> I thought there would be an exponential version of the CLT, as
> many varying exponentials.
>
> bc  wrong

Exponential version of the CLT? The CLT requires only a large sum of independent random variables each with a finite mean and finite standard deviation, and with bounds on the distribution of those means and standard deviations of the random variables in the sum.  The individual random variables in the sum can have any shape as long as they each have a finite mean and finite variance.  Of course exponentials with positive exponents don't have any finite moments at all, and they are not even normalizable, anyway (unless one restricts them to a finite domain to the right, in which case then all the moments become finite and the ordinary CLT would then apply).

What I find remarkable about the statistics of this pandemic is the unreasonable effectiveness of the cumulative *log-normal* family of model distributions in fitting the cumulative death data for the US & some other countries.  There doesn't seem to be a simple theoretical explanation of the success of cumulative log-normal models.  We *would* expect to find a log-normal distribution when the modeled random variable is expressible as a *product* of a huge number of independent random positive factors, each of which have finite and bounded first 2 moments of their *logarithms*.  A log-normal distribution is also the minimal entropy distribution for a positive random variable which has given fixed values for the first 2 moments of its logarithm (or its geometric mean & geometric mean square).  However I confess I don't see how any of this should apply to the *time series* of the daily death for this pandemic.

Dave Bowman