# Fractional Divergence of Probability Densities

## Abstract

**:**

## 1. Introduction

## 2. Divergence between Two Probability Densities

## 3. Conventional Divergence of Exponential and Pareto Densities

## 4. Fractional Divergence of Exponential and Pareto Densities

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

## 5. Manipulation of the Divergence between Two Exponential Densities via the Fractional Orders

## 6. An Application of the Fractional Divergence to Detection Theory

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The divergence between the Exponential-density and the Pareto-density for a fixed Pareto scale parameter, ${x}_{0}=0.01$.

**Figure 2.**For ${x}_{0}=0.01$, the (

**a**) conventional and (

**b**) fractional divergence is shown, respectively.

**Figure 3.**Variation of the (fractional) divergence manifold between two Exponential-densities in terms of the fractional orders ${\alpha}_{1}$ and ${\alpha}_{2}$. The case $\alpha =1$ corresponds to the conventional divergence.

**Figure 4.**Further manipulation of the (fractional) divergence manifold between two Exponential-densities via the fractional orders ${\alpha}_{1}$ and ${\alpha}_{2}$.

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Alexopoulos, A.
Fractional Divergence of Probability Densities. *Fractal Fract.* **2017**, *1*, 8.
https://doi.org/10.3390/fractalfract1010008

**AMA Style**

Alexopoulos A.
Fractional Divergence of Probability Densities. *Fractal and Fractional*. 2017; 1(1):8.
https://doi.org/10.3390/fractalfract1010008

**Chicago/Turabian Style**

Alexopoulos, Aris.
2017. "Fractional Divergence of Probability Densities" *Fractal and Fractional* 1, no. 1: 8.
https://doi.org/10.3390/fractalfract1010008