Probability density distribution function for the maximum information entropy

7 Probability density distribution function for the maximum information entropy 

It has been said in the natural world, the aim is to achieve the maximum value of information entropy. In this section, the relationship between a probability density distribution function and the maximum value of the information entropy is discussed. In the case of a mathematical discussion, it is easier to treat information entropy H(t) based on continuous variables H(t) rather than the information entropy based on discrete variables H X). In the following, H(t) is studied, and the probability density distribution function p(t) for the maximum value of information entropy H(t)max under three typical restriction conditions is shown. 

The standardized condition of the probability density distribution function is given as 

The range of integration shows the given restrictive condition. Under these conditions, the form of the probability density distribution function p(t) for the maximum value of information entropy is investigated. By using calculus 

The calculus of variations is applied to H(i) by introducing the variable coefficient A. When the variation method is applied to H(t) as 

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