# Distribution Function, Probability Generating Function and Archimedean Generator

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Additive and Multiplicative Archimedean Generators

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**2.**

## 3. Distribution Functions as Archimedean Generators

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

**Corollary**

**1.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 4. Probability Generating Functions as Multiplicative Archimedean Generators

**Definition**

**5.**

**Theorem**

**6.**

**Example**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**7.**

**Proof.**

**Remark**

**1.**

**Theorem**

**8.**

**Proof.**

- Continuous on I.
- Strictly increasing on I.
- ${G}^{-1}\left(1\right)=1$ and ${G}^{-1}\left(0\right)=0$.
- Log-concave. Because we have $\frac{d}{du}ln\left[{G}^{-1}\left(u\right)\right]=\frac{1}{{G}^{{}^{\prime}}\left[{G}^{-1}\left(u\right)\right]{G}^{-1}\left(u\right)}$ decreasing. Since both ${G}^{-1}\left(u\right)$ and ${G}^{{}^{\prime}}\left[{G}^{-1}\left(u\right)\right]$ are increasing.

**Example**

**6.**

**Corollary**

**2.**

**Example**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Definition**

**A1.**

**Theorem**

**A1**

**.**Suppose $K(x,y)>0$, is $T{P}_{r}$ function and $f\left(y\right)$ changes sign $j\le r-1$ times. Let

**Proof**

**of Theorem 7.**

**Table A1.**Some copulas generated by survival functions $\phantom{\rule{0.83328pt}{0ex}}\overline{\phantom{\rule{-0.83328pt}{0ex}}F\phantom{\rule{-0.83328pt}{0ex}}}\phantom{\rule{0.83328pt}{0ex}}$.

Distribution | Generator | Parameter | Copula |
---|---|---|---|

Beta$(1,\beta )$ | ${(1-x)}^{\beta}$ | $\beta <1$ | $max\{0,1-{\left[{(1-u)}^{\beta}+{(1-v)}^{\beta}\right]}^{\frac{1}{\beta}}\}$ |

Beta$(\alpha ,1)$ | $1-{x}^{\alpha}$ | $\alpha <1$ | $max\{0,{({u}^{\alpha}+{v}^{\alpha}-1)}^{\frac{1}{\alpha}}\}$ |

Truncated BetaPrim$(1,\beta )$ | $\frac{{(1+x)}^{-\beta}-{2}^{-\beta}}{1-{2}^{-\beta}}$ | $\beta >0$ | $max\{0,{\left[{(1+u)}^{-\beta}+{(1+v)}^{-\beta}-{2}^{-\beta}\right]}^{-\frac{1}{\beta}}-1\}$ |

Exponential-Logarithmic$(1,p)$ | $\frac{ln\left(\frac{1-q{e}^{-x}}{1-q{e}^{-1}}\right)}{ln\left(\frac{p}{1-q{e}^{-1}}\right)}$ | $0<p<1,$$q=1-p$ | $max\{0,-ln\left[\frac{1}{q}+\frac{(q{e}^{-u}-1)(q{e}^{-v}-1)}{q(q{e}^{-1}-1)}\right]\}$ |

**Table A2.**Some copulas generated by $\phantom{\rule{0.83328pt}{0ex}}\overline{\phantom{\rule{-0.83328pt}{0ex}}H\phantom{\rule{-0.83328pt}{0ex}}}\phantom{\rule{0.83328pt}{0ex}}\left(x\right)=F(1-x)$.

Distribution | Generator | Parameter | Copula |
---|---|---|---|

Beta$(1,\beta )$ | $1-{x}^{\beta}$ | $\beta >1$ | $max\{0,{({u}^{\beta}+{v}^{\beta}-1)}^{\frac{1}{\beta}}\}$ |

Beta$(\alpha ,1)$ | ${(1-x)}^{\alpha}$ | $\alpha >1$ | $max\{0,1-{\left[{(1-u)}^{\alpha}+{(1-v)}^{\alpha}\right]}^{\frac{1}{\alpha}}\}$ |

Distribution | Generator | Parameter | Copula |
---|---|---|---|

Beta$(\alpha ,1)$ | ${x}^{\frac{1}{\alpha}}$ | $\alpha >0$ | $uv$ |

Lomax$(1,\beta )$ | $\frac{\beta x}{1+\beta -x}$ | $\beta >1$ | $\frac{uv}{1+\frac{1}{\beta}(1-u)(1-v)}$ |

Distribution | Generator | Parameter | Copula |
---|---|---|---|

Binomial$(n,p)$ | $G\left(s\right)={(1-p+ps)}^{n}$ | $0<p<1$ | $max\left\{\right(1-p\left)\right(u+v-1)+puv,0\}$ |

Geometric$\left(p\right)$ | $G\left(s\right)=\frac{ps}{1-(1-p)s}$ | $0<p<1$ | $\frac{uv}{1+\frac{1-p}{p}(1-u)(1-v)}$ |

Poisson$\left(\mu \right)$ | $G\left(s\right)={e}^{\mu (s-1)}$ | $\mu >0$ | $max\{u+v-1,0\}$ |

Truncated Poisson$\left(\mu \right)$ | $G\left(s\right)=\frac{{e}^{\mu *s}-1}{{e}^{\mu}-1}$ | $\mu >0$ | $\frac{1}{\mu}ln[1+\frac{({e}^{\mu u}-1)({e}^{\mu v}-1)}{{e}^{\mu}-1}]$ |

Pascal$\left(p\right)$ | ${G}^{-1}\left(s\right)={\left[\frac{(1-p)s}{1-ps}\right]}^{r}$ | $0<p<1$$r:$ Integer | $\frac{uv}{{\left[1-p\left(1-{u}^{\frac{1}{r}}\right)\left(1-{v}^{\frac{1}{r}}\right)\right]}^{r}}$ |

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Alhadlaq, W.; Alzaid, A.
Distribution Function, Probability Generating Function and Archimedean Generator. *Symmetry* **2020**, *12*, 2108.
https://doi.org/10.3390/sym12122108

**AMA Style**

Alhadlaq W, Alzaid A.
Distribution Function, Probability Generating Function and Archimedean Generator. *Symmetry*. 2020; 12(12):2108.
https://doi.org/10.3390/sym12122108

**Chicago/Turabian Style**

Alhadlaq, Weaam, and Abdulhamid Alzaid.
2020. "Distribution Function, Probability Generating Function and Archimedean Generator" *Symmetry* 12, no. 12: 2108.
https://doi.org/10.3390/sym12122108