Information entropy based on continuous variable

Until now, the variable in every system has been discrete for example, the number obtained when a die is cast. As not only a discrete variable but also a continuous variable (for example, time) appears very often in chemical engineering, it is necessary to define the information entropy for continuous variable. Of course, it is possible to define the average amount of information for a system that is based on a continuous variable, for example, time. In the case based on a continuous variable t, let p tt) and At be the probability density at t and the very small change of continuous variable, respectively. The product of pif) and At corresponds to the probability P , in Eq. (1.3). When this method of 

The second term on the right-hand side of this equation takes an infinite value regardless of the values of p(t), and only the first term changes in response to the change in the probability density distribution function p(t). Therefore, the information entropy based on the continuous variable is defined as 

The information entropy defined by Eq. (1.10) is as important as that defined by Eq. (1.3) in chemical engineering because there are a number of phenomena that are controlled by time as a variable in other fields of engineering as well. 

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