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

​The 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 Read more…

## Long hand division generation of polynomials

Do a long hand division of   x greater than or equal to 1 does not result in convergence of this sum.  However this algorithm can still be used to do some interesting things.  Let us use a complex value of    Each power of x yields a result one step around Read more…

## Z transform of and exponentially decaying sequence

The series:          this converges for x < 1 : Both of these expressions are the Z transform of the  exponential decay sequence.  The first expression is easier to deal with because it is smaller and easier to work with.  The following diagram uses a decay sequence with   The filter Read more…

## Two 2 dimensional determinant of a matrix animation showing it is equal to the area of the parallelogram

The 2 dimensional determinant of a matrix can be interpreted as the area of a parallelogram as shown in the following diagram. This carries on through higher dimensions.  Below depicts a 3 variable system. The rows r1, r2, r3 are vectors each. The various summations taken 1, 2 and 3 Read more…

## Derivation of the Normal Gaussian distribution from physical principles

In many physical systems the question arises what is the probability distribution that describes a system with a given expected energy E  over the interval from -infinity to + infinity?     Again you will use the maximum entropy principle to determine this. The constraints are as follows:         …. sum over Read more…

## The Maximum Entropy Principle – The distribution with the maximum entropy is the distribution nature chooses

In a previous article entropy was defined as the expected number of bits in a binary number required to enumerate all the outcomes.  This was expressed as follows: entropy= H(x)=   In physics ( nature ) it is found that the probability distribution that represents a physical process is the Read more…

## Use of Maximum Entropy to explain the form of Energy States of an Electron in a Potential Well

The base state of an electron in an infinite potential well has the most "space" for the electron state.  Thus it has the maximum entropy. Take that same state and imagine pinching the electrons existence to nil in the middle of the trough.  Now you have state-2.  The electron now Read more…