### Maximum Entropy Distribution for Random Variable of Extent [0,Infinity] and a Mean Value Mu

Sunday, June 22nd, 2014The maximum entropy constraints are as follows:

- Over the interval [0,infinity]
- …. sum over all probabities must = 1
- …. given an average value AKA "mean"

The langrangian is formed as follows:

….setting equal to zero to find the extrema point

Allowing to take up the slack to turn base 2 log into natural log:

Using the sum of probabilities =1 criteria

( See below for derivation)

**Derivation of mean value infinite sum:**

Subtracting we get the same old geometric series that we all know

Rearranging terms:

Another way of looking at the series:

Infinite Series Multiplication Table – The product of the 2 series is the sum of all the product entries ad infinitum | ||||

… | … | … | ||

… | … | |||

… | ||||

The table uses 2 exponential series each starting with 1. In order to get the same series as the solution in the derivation above multiple the result by

It forms a sort of number wedge or number cone. I wonder if it extends to 3 dimensions?

__ Observations__ ( Need to complete this )

- ….delay like Z transform
- continuous form correspondence with discrete form

**Research Links**