# Some Inequalities Combining Rough and Random Information

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Random Variable

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1.**

#### 2.2. Rough Variable

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Example**

**1.**

**Definition**

**8.**

**Example**

**2.**

**Theorem**

**2.**

#### 2.3. Rough Random Variable

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**3.**

**Definition**

**11.**

**Definition**

**12.**

## 3. Inequalities of Rough Random Variables

**Theorem**

**4.**

**Proof of Theorem**

**4.**

**Theorem**

**5.**

**Proof of Theorem**

**5.**

**Theorem**

**6.**

**Proof of Theorem**

**6.**

**Theorem**

**7.**

**Proof of Theorem**

**7.**

**Theorem**

**8.**

**Proof of Theorem**

**8.**

**Theorem**

**9.**

**Proof of Theorem**

**9.**

**Theorem**

**10.**

**Proof of Theorem**

**10.**

**Theorem**

**11.**

## 4. Critical Values of Rough Random Variables

**Definition**

**13.**

**Theorem**

**12.**

**Theorem**

**13.**

**Proof of Theorem**

**13.**

**Theorem**

**14.**

**Proof of Theorem**

**14.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Gu, Y.; Zhang, Q.; Yu, L.
Some Inequalities Combining Rough and Random Information. *Entropy* **2018**, *20*, 211.
https://doi.org/10.3390/e20030211

**AMA Style**

Gu Y, Zhang Q, Yu L.
Some Inequalities Combining Rough and Random Information. *Entropy*. 2018; 20(3):211.
https://doi.org/10.3390/e20030211

**Chicago/Turabian Style**

Gu, Yujie, Qianyu Zhang, and Liying Yu.
2018. "Some Inequalities Combining Rough and Random Information" *Entropy* 20, no. 3: 211.
https://doi.org/10.3390/e20030211