# The Uniform Poisson–Ailamujia Distribution: Actuarial Measures and Applications in Biological Science

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## Abstract

**:**

## 1. Introduction

## 2. The Discrete UPA Distribution

#### 2.1. Properties

#### 2.2. Stochastic Orders of the Parameter $\alpha $

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

#### 2.3. Entropy

#### 2.4. Quantile Function

## 3. Actuarial Measures

#### 3.1. VaR Measure

#### 3.2. TVaR Measure

## 4. Estimation

#### 4.1. Maximum Likelihood

#### 4.2. Moments

#### 4.3. Proportions

#### 4.4. Ordinary and Weighted Least-Squares

#### 4.5. Cramér-von Mises

#### 4.6. Right-Tail Anderson–Darling

#### 4.7. Percentiles

## 5. Simulation Study

`R`software.

## 6. Modeling Biological Data

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Continuous Distribution | Discrete Distribution | Author |
---|---|---|

Weibull | Discrete Weibull | Nakagawa and Osaki [1] |

Inverse Weibull | Discrete inverse Weibull | Stein and Dattero [2] |

Normal and Rayleigh | Discrete normal and Rayleigh | Roy [3,4] |

Burr XII and Pareto | Discrete Burr XII and Pareto | Krishna and Pundir [5] |

Gamma | Discrete gamma | Chakraborty and Chakravarty [6] |

Chen | Discrete Chen | Noughabi et al. [7] |

$\mathit{\alpha}$ | Mean | B | $\mathit{E}\left({\mathit{X}}^{2}\right)$ | $\mathit{E}\left({\mathit{X}}^{3}\right)$ | $\mathit{E}\left({\mathit{X}}^{4}\right)$ | ID |
---|---|---|---|---|---|---|

0.25 | 2.0000 | 6.0000 | 10.0000 | 74.0000 | 730.0000 | 3.0000 |

0.75 | 0.6667 | 1.1111 | 1.5555 | 5.1111 | 22.2963 | 1.6667 |

1.00 | 0.5000 | 0.7500 | 1.0000 | 2.7500 | 10.0000 | 1.5000 |

1.25 | 0.4000 | 0.5600 | 0.7200 | 1.7440 | 5.5584 | 1.4000 |

1.50 | 0.3333 | 0.4444 | 0.5555 | 1.2222 | 3.5185 | 1.3333 |

1.75 | 0.2857 | 0.3673 | 0.4490 | 0.9154 | 2.4281 | 1.2857 |

2.00 | 0.2500 | 0.3125 | 0.3750 | 0.7187 | 1.7812 | 1.2500 |

2.25 | 0.2222 | 0.2716 | 0.3210 | 0.5844 | 1.3672 | 1.2222 |

2.50 | 0.2000 | 0.2400 | 0.2800 | 0.4880 | 1.0864 | 1.2000 |

2.75 | 0.1818 | 0.2149 | 0.2479 | 0.4162 | 0.8872 | 1.1818 |

3.25 | 0.1538 | 0.1775 | 0.2011 | 0.3177 | 0.6297 | 1.1538 |

3.75 | 0.1333 | 0.1511 | 0.1689 | 0.2542 | 0.4751 | 1.1333 |

4.50 | 0.1111 | 0.1234 | 0.1358 | 0.1934 | 0.3370 | 1.1111 |

5.50 | 0.0909 | 0.0992 | 0.1074 | 0.1450 | 0.2353 | 1.0909 |

7.50 | 0.0667 | 0.0711 | 0.0755 | 0.0951 | 0.1400 | 1.0667 |

9.50 | 0.0526 | 0.0554 | 0.0582 | 0.0701 | 0.0968 | 1.0526 |

10.00 | 0.0500 | 0.0525 | 0.0550 | 0.0657 | 0.0896 | 1.0500 |

50.00 | 0.0100 | 0.0101 | 0.0102 | 0.0106 | 0.0114 | 1.0100 |

75.00 | 0.0067 | 0.0067 | 0.0067 | 0.0069 | 0.0073 | 1.0067 |

100.00 | 0.0050 | 0.0050 | 0.0050 | 0.0051 | 0.0053 | 1.0050 |

$\mathit{\alpha}$ | $\mathit{H}\left(\mathit{X}\right)$ | $\mathit{\alpha}$ | $\mathit{H}\left(\mathit{X}\right)$ | $\mathit{\alpha}$ | $\mathit{H}\left(\mathit{X}\right)$ |
---|---|---|---|---|---|

3.5 | 0.4306 | 7 | 0.2624 | ||

0.5 | 1.3863 | 4 | 0.3924 | 7.5 | 0.2494 |

1 | 0.9548 | 4.5 | 0.3612 | 8 | 0.2377 |

1.5 | 0.7498 | 5 | 0.3351 | 8.5 | 0.2272 |

2 | 0.6255 | 5.5 | 0.3129 | 9 | 0.2176 |

2.5 | 0.5407 | 6 | 0.2938 | 9.5 | 0.2090 |

3 | 0.4785 | 6.5 | 0.2771 | 10 | 0.2010 |

$\mathit{\alpha}$ | Security Level | VaR${}_{\mathit{p}}$ | TVaR${}_{\mathit{p}}$ | |||
---|---|---|---|---|---|---|

0.25 | 0.80 | 2.9694 | 7.4074 | |||

0.85 | 3.6789 | 7.9012 | ||||

0.90 | 4.6789 | 9.2181 | ||||

0.95 | 6.3884 | 10.5350 | ||||

0.99 | 10.3577 | 15.0293 | ||||

0.5 | 0.80 | 1.3219 | 4.6338 | |||

0.85 | 1.7369 | 5.4739 | ||||

0.90 | 2.3219 | 6.6438 | ||||

0.95 | 3.3219 | 8.6438 | ||||

0.99 | 5.6438 | 13.2877 | ||||

1.5 | 0.80 | 0.1610 | 1.9772 | |||

0.85 | 0.3685 | 2.8073 | ||||

0.90 | 0.6610 | 3.9772 | ||||

0.95 | 1.1610 | 5.9772 | ||||

0.99 | 2.3219 | 10.6210 |

n | MLE | POE | MOE | LSE | WLSE | CVME | RADE | PCE | |
---|---|---|---|---|---|---|---|---|---|

30 | AVEs | 0.3571 | 0.3333 | 0.3571 | 0.3633 | 0.3635 | 0.3754 | 0.3742 | 0.3724 |

MSEs | 0.3031 | 0.0076 | 0.0031 | 0.0039 | 0.0040 | 0.0205 | 0.0198 | 0.0764 | |

ABSs | 0.0554 | 0.0875 | 0.0554 | 0.0626 | 0.0631 | 0.0863 | 0.0789 | 0.2624 | |

MREs | 0.1583 | 0.2504 | 0.1583 | 0.1788 | 0.1804 | 0.2415 | 0.1428 | 0.8639 | |

75 | AVEs | 0.3505 | 0.3523 | 0.3505 | 0.3555 | 0.3557 | 0.3495 | 0.3465 | 0.3639 |

MSEs | 0.0013 | 0.0027 | 0.0013 | 0.0017 | 0.0018 | 0.0101 | 0.0087 | 0.0458 | |

ABSs | 0.0366 | 0.0521 | 0.0366 | 0.0415 | 0.0421 | 0.0774 | 0.0712 | 0.2139 | |

MREs | 0.1046 | 0.1489 | 0.1046 | 0.1184 | 0.1204 | 0.2212 | 0.0712 | 0.6112 | |

100 | AVEs | 0.3521 | 0.3474 | 0.3521 | 0.3525 | 0.3521 | 0.3394 | 0.3416 | 0.3480 |

MSEs | 0.0010 | 0.0019 | 0.0010 | 0.0013 | 0.0013 | 0.0057 | 0.0050 | 0.0065 | |

ABSs | 0.0315 | 0.0435 | 0.0315 | 0.0355 | 0.0360 | 0.0602 | 0.0557 | 0.0634 | |

MREs | 0.0901 | 0.1244 | 0.0908 | 0.1017 | 0.1029 | 0.1719 | 0.0557 | 0.1812 | |

150 | AVEs | 0.3505 | 0.3523 | 0.3505 | 0.3524 | 0.3524 | 0.3397 | 0.3403 | 0.3478 |

MSEs | 0.0006 | 0.0018 | 0.0006 | 0.0008 | 0.0008 | 0.0045 | 0.0039 | 0.0049 | |

ABSs | 0.0250 | 0.0428 | 0.0250 | 0.0291 | 0.0294 | 0.0537 | 0.0498 | 0.0556 | |

MREs | 0.0714 | 0.1224 | 0.0714 | 0.0832 | 0.0841 | 0.1535 | 0.0498 | 0.1589 | |

200 | AVEs | 0.3509 | 0.3474 | 0.3508 | 0.3525 | 0.3525 | 0.3354 | 0.3389 | 0.3418 |

MSEs | 0.0005 | 0.0012 | 0.0005 | 0.0006 | 0.0006 | 0.0023 | 0.0020 | 0.0024 | |

ABSs | 0.0217 | 0.0349 | 0.0217 | 0.0247 | 0.0248 | 0.0392 | 0.0360 | 0.0393 | |

MREs | 0.0621 | 0.0999 | 0.0621 | 0.0707 | 0.0710 | 0.1121 | 0.0360 | 0.1123 | |

300 | AVEs | 0.3505 | 0.3522 | 0.3505 | 0.3519 | 0.3516 | 0.3463 | 0.3465 | 0.3488 |

MSEs | 0.0003 | 0.0007 | 0.0003 | 0.0004 | 0.0004 | 0.0010 | 0.0008 | 0.0009 | |

ABSs | 0.0176 | 0.0272 | 0.0176 | 0.0203 | 0.0204 | 0.0261 | 0.0226 | 0.0236 | |

MREs | 0.0504 | 0.0777 | 0.0504 | 0.0579 | 0.0583 | 0.0745 | 0.0226 | 0.0673 |

n | MLE | POE | MOE | LSE | WLSE | CVME | RADE | PCE | |
---|---|---|---|---|---|---|---|---|---|

30 | AVEs | 0.5172 | 0.5127 | 0.5172 | 0.5146 | 0.5155 | 0.4757 | 0.4785 | 0.4898 |

MSEs | 0.0069 | 0.0138 | 0.0069 | 0.0089 | 0.0089 | 0.0167 | 0.0147 | 0.0217 | |

ABBs | 0.0833 | 0.1176 | 0.0833 | 0.0943 | 0.0944 | 0.1028 | 0.0967 | 0.1136 | |

MREs | 0.1667 | 0.2353 | 0.1667 | 0.1886 | 0.1888 | 0.2050 | 0.1934 | 0.2273 | |

75 | AVEs | 0.5145 | 0.4868 | 0.5145 | 0.5020 | 0.5016 | 0.4653 | 0.4713 | 0.4848 |

MSEs | 0.0029 | 0.0051 | 0.0029 | 0.0037 | 0.0038 | 0.0073 | 0.0060 | 0.0077 | |

ABBs | 0.0536 | 0.0714 | 0.0536 | 0.0612 | 0.0615 | 0.0703 | 0.0634 | 0.0704 | |

MREs | 0.1071 | 0.1429 | 0.1071 | 0.1224 | 0.1230 | 0.1407 | 0.1267 | 0.1408 | |

100 | AVEs | 0.5126 | 0.4868 | 0.5127 | 0.5020 | 0.5016 | 0.4619 | 0.4588 | 0.4871 |

MSEs | 0.0024 | 0.0041 | 0.0024 | 0.0029 | 0.0029 | 0.0056 | 0.0045 | 0.0059 | |

ABBs | 0.0495 | 0.0638 | 0.0495 | 0.0542 | 0.0542 | 0.0620 | 0.0562 | 0.0615 | |

MREs | 0.0989 | 0.1277 | 0.0989 | 0.1085 | 0.1085 | 0.1241 | 0.1124 | 0.1230 | |

150 | AVEs | 0.5098 | 0.5057 | 0.5097 | 0.5023 | 0.5019 | 0.4588 | 0.4678 | 0.4894 |

MSEs | 0.0016 | 0.0032 | 0.0016 | 0.0020 | 0.0019 | 0.0045 | 0.0036 | 0.0040 | |

ABBs | 0.0396 | 0.0563 | 0.0396 | 0.0447 | 0.0441 | 0.0562 | 0.0497 | 0.0508 | |

MREs | 0.0791 | 0.1127 | 0.0791 | 0.0894 | 0.883 | 0.1124 | 0.0993 | 0.1016 | |

200 | AVEs | 0.5044 | 0.5005 | 0.5045 | 0.5011 | 0.5008 | 0.4600 | 0.4665 | 0.4906 |

MSEs | 0.0011 | 0.0023 | 0.0011 | 0.0014 | 0.0014 | 0.0037 | 0.0030 | 0.0030 | |

ABBs | 0.0327 | 0.0476 | 0.0327 | 0.0379 | 0.0379 | 0.0510 | 0.0456 | 0.0442 | |

MREs | 0.0654 | 0.0952 | 0.0654 | 0.0755 | 0.0758 | 0.1021 | 0.0911 | 0.0883 | |

300 | AVEs | 0.5005 | 0.5004 | 0.5004 | 0.5003 | 0.5003 | 0.4573 | 0.4667 | 0.4914 |

MSEs | 0.0008 | 0.0015 | 0.0008 | 0.0010 | 0.0010 | 0.0031 | 0.0024 | 0.0019 | |

ABBs | 0.0282 | 0.0385 | 0.0282 | 0.0312 | 0.0315 | 0.0480 | 0.0406 | 0.0354 | |

MREs | 0.0563 | 0.0769 | 0.0563 | 0.0625 | 0.0630 | 0.0961 | 0.0813 | 0.0708 |

n | MLE | POE | MOE | LSE | WLSE | CVME | RADE | PCE | |
---|---|---|---|---|---|---|---|---|---|

30 | AVEs | 1.5421 | 1.6429 | 1.5521 | 1.6069 | 1.6044 | 1.1887 | 1.2015 | 1.5306 |

MSEs | 0.1406 | 0.2500 | 0.1406 | 0.2060 | 0.2029 | 0.2316 | 0.2203 | 1.0837 | |

ABBs | 0.3750 | 0.5024 | 0.3750 | 0.4538 | 0.4504 | 0.4160 | 0.4045 | 0.4863 | |

MREs | 0.2500 | 0.3333 | 0.2501 | 0.3025 | 0.3003 | 0.2773 | 0.2696 | 0.3241 | |

75 | AVEs | 1.5215 | 1.4737 | 1.5257 | 1.4992 | 1.4941 | 1.1389 | 1.1612 | 1.4721 |

MSEs | 0.0428 | 0.0873 | 0.0428 | 0.0604 | 0.0617 | 0.1750 | 0.1594 | 0.1434 | |

ABBs | 0.2069 | 0.2955 | 0.2069 | 0.2457 | 0.2483 | 0.3780 | 0.3598 | 0.2931 | |

MREs | 0.1379 | 0.1970 | 0.1379 | 0.1638 | 0.1656 | 0.2520 | 0.2399 | 0.1954 | |

100 | AVEs | 1.5152 | 1.5247 | 1.5152 | 1.5029 | 1.5028 | 1.1337 | 1.1641 | 1.4760 |

MSEs | 0.0475 | 0.0459 | 0.0475 | 0.0469 | 0.0470 | 0.1676 | 0.1450 | 0.1075 | |

ABBs | 0.2179 | 0.2143 | 0.2179 | 0.2166 | 0.2169 | 0.3748 | 0.3471 | 0.2555 | |

MREs | 0.1453 | 0.1429 | 0.1453 | 0.1444 | 0.1446 | 0.2499 | 0.2314 | 0.1703 | |

150 | AVEs | 1.5158 | 1.5270 | 1.5247 | 1.5110 | 1.5120 | 1.1305 | 1.1600 | 1.4791 |

MSEs | 0.0278 | 0.0424 | 0.0278 | 0.0348 | 0.0347 | 0.1575 | 0.1364 | 0.0069 | |

ABBs | 0.1667 | 0.2059 | 0.1667 | 0.1865 | 0.1862 | 0.3714 | 0.3431 | 0.2073 | |

MREs | 0.1111 | 0.1373 | 0.1111 | 0.1243 | 0.1241 | 0.2476 | 0.2288 | 01382 | |

200 | AVEs | 1.5152 | 1.5123 | 1.5152 | 1.5000 | 1.4995 | 1.1211 | 1.1522 | 1.4794 |

MSEs | 0.0221 | 0.0302 | 0.0220 | 0.0265 | 0.0265 | 0.1590 | 0.1306 | 0.0494 | |

ABBs | 0.1486 | 0.1739 | 0.1486 | 0.1627 | 0.1629 | 0.3795 | 0.3479 | 0.1778 | |

MREs | 0.0991 | 0.1159 | 0.0991 | 0.1084 | 0.1086 | 0.2530 | 0.2330 | 0.1186 | |

300 | AVEs | 1.5045 | 1.5057 | 1.5098 | 1.5032 | 1.5028 | 1.1205 | 1.1522 | 1.4813 |

MSEs | 0.0127 | 0.0156 | 0.0127 | 0.0160 | 0.0159 | 0.1543 | 0.1306 | 0.0329 | |

ABBs | 0.1129 | 0.1250 | 0.1129 | 0.1265 | 0.1260 | 0.3766 | 0.3479 | 0.1452 | |

MREs | 0.0753 | 0.0833 | 0.0752 | 0.0844 | 0.0840 | 0.2531 | 0.2319 | 0.0968 |

n | MLE | POE | MOE | LSE | WLSE | CVME | RADE | PCE | |
---|---|---|---|---|---|---|---|---|---|

30 | AVEs | 3.4270 | 3.2500 | 3.2347 | 3.2366 | 3.2866 | 2.5001 | 2.5030 | 3.4441 |

MSEs | 0.7347 | 1.0124 | 0.7347 | 1.0417 | 1.0248 | 1.5788 | 1.5904 | 1.8639 | |

ABBs | 0.8571 | 1.0147 | 0.8571 | 1.0206 | 1.0123 | 1.1359 | 1.1275 | 1.4804 | |

MREs | 0.2857 | 0.3333 | 0.2857 | 0.3402 | 0.3374 | 0.3786 | 0.3758 | 0.4935 | |

75 | AVEs | 3.1250 | 2.9091 | 3.1250 | 2.9339 | 2.9364 | 2.8859 | 2.9132 | 3.0907 |

MSEs | 0.4307 | 0.4444 | 0.4307 | 0.4302 | 0.4304 | 1.4055 | 1.3384 | 1.2770 | |

ABBs | 0.6562 | 0.6667 | 0.6563 | 0.6560 | 0.6561 | 1.1236 | 1.0963 | 0.7931 | |

MREs | 0.2188 | 0.2222 | 0.2188 | 0.2187 | 0.2187 | 0.3745 | 0.3654 | 0.2644 | |

100 | AVEs | 3.1250 | 3.0714 | 3.1250 | 3.0711 | 3.0711 | 2.8642 | 2.8988 | 3.0839 |

MSEs | 0.3265 | 0.3122 | 0.3265 | 0.3318 | 0.3289 | 1.4062 | 1.3243 | 0.8779 | |

ABBs | 0.5714 | 0.5588 | 0.5714 | 0.5760 | 0.5735 | 1.1388 | 1.1045 | 0.6850 | |

MREs | 0.1905 | 0.1863 | 0.1905 | 0.1920 | 0.1912 | 0.3796 | 0.3681 | 0.2283 | |

150 | AVEs | 3.0785 | 3.0714 | 3.0673 | 3.0650 | 3.0656 | 2.8513 | 2.8892 | 3.0408 |

MSEs | 0.1712 | 0.2001 | 0.1712 | 0.2026 | 0.2031 | 1.3906 | 1.3021 | 0.5301 | |

ABBs | 0.4138 | 0.4474 | 0.4138 | 0.4502 | 0.4507 | 1.1487 | 1.1109 | 0.5437 | |

MREs | 0.1379 | 0.1491 | 0.1379 | 0.1500 | 0.1502 | 0.3829 | 0.3703 | 0.1819 | |

200 | AVEs | 3.0303 | 3.0714 | 3.0303 | 3.0578 | 3.0562 | 2.8470 | 2.8804 | 3.0547 |

MSEs | 0.1357 | 0.1406 | 0.1357 | 0.1439 | 0.1449 | 1.3820 | 1.3047 | 0.3779 | |

ABBs | 0.3684 | 0.3750 | 0.3684 | 0.3794 | 0.3807 | 1.1530 | 1.1197 | 0.4726 | |

MREs | 0.1228 | 0.1250 | 0.1228 | 0.1265 | 0.1269 | 0.3844 | 0.3732 | 0.1576 | |

300 | AVEs | 3.0159 | 2.9884 | 3.0158 | 2.9923 | 2.9917 | 2.8412 | 2.8765 | 3.0507 |

MSEs | 0.1033 | 0.1198 | 0.1033 | 0.1178 | 0.1183 | 1.3785 | 1.2945 | 0.2502 | |

ABBs | 0.3214 | 0.3462 | 0.3214 | 0.3432 | 0.3439 | 1.1588 | 1.1235 | 0.3854 | |

MREs | 0.1071 | 0.1154 | 0.1071 | 0.1144 | 0.1146 | 0.3863 | 0.3745 | 0.1285 |

Count | Observed | Expected | ||||||
---|---|---|---|---|---|---|---|---|

UPA | DBH | NDL | Poisson | Pareto | PA | DPL | ||

0 | 268 | 264.03 | 282.33 | 258.76 | 238.99 | 292.41 | 252.96 | 262.44 |

1 | 87 | 89.75 | 71.52 | 95.87 | 123.09 | 57.68 | 103.60 | 91.61 |

2 | 26 | 30.51 | 25.79 | 31.57 | 31.70 | 20.97 | 31.82 | 30.92 |

3 | 9 | 10.37 | 10.78 | 9.75 | 5.44 | 9.98 | 8.69 | 10.19 |

4 | 10 | 3.53 | 4.89 | 2.89 | 0.70 | 5.54 | 2.22 | 3.30 |

Total | $n=$ 400 | |||||||

Parameters | $\widehat{\alpha}$ | 0.9709 | 0.5883 | 0.7530 | 0.5150 | 0.1504 | 1.9417 | 2.5012 |

${\chi}^{2}$ | 5.33 | 6.49 | 7.54 | 29.41 | 15.17 | 11.37 | 5.94 | |

$-\widehat{\ell}$ | 388.44 | 389.76 | 390.99 | 408.63 | 405.12 | 392.55 | 388.74 |

Count | Observed | Expected | ||||||
---|---|---|---|---|---|---|---|---|

UPA | DBH | NDL | Poisson | Pareto | PA | DPL | ||

0 | 200 | 195.80 | 209.55 | 190.60 | 174.83 | 216.88 | 186.00 | 193.45 |

1 | 57 | 68.31 | 54.09 | 73.07 | 94.40 | 43.89 | 79.09 | 69.82 |

2 | 30 | 24.96 | 19.92 | 24.90 | 25.49 | 16.20 | 25.22 | 24.34 |

3 | 7 | 8.41 | 8.51 | 7.96 | 4.59 | 7.80 | 7.15 | 8.28 |

4≥ | 6 | 3.06 | 3.95 | 2.44 | 0.62 | 4.37 | 1.90 | 2.76 |

Total | $n=$ 300 | |||||||

Parameters | $\widehat{\alpha}$ | 0.9259 | 0.6030 | 0.7444 | 0.5400 | 0.1570 | 1.8518 | 2.4002 |

${\chi}^{2}$ | 4.90 | 5.02 | 8.08 | 34.04 | 11.32 | 13.10 | 6.15 | |

$-\widehat{\ell}$ | 299.31 | 301.70 | 300.16 | 314.23 | 312.94 | 302.41 | 302.41 |

Model | $\widehat{\mathit{\alpha}}$ | $\mathbf{AIC}$ | $\mathbf{BIC}$ | $-\mathit{\ell}$ |
---|---|---|---|---|

UPA | 0.0585 | 757.2386 | 757.3214 | 377.6193 |

DPL | 0.2318 | 765.8140 | 765.8968 | 381.9070 |

NDL | 0.1901 | 767.1740 | 767.2568 | 382.5870 |

PA | 0.1275 | 778.3318 | 778.4146 | 388.1659 |

DP | 0.5857 | 849.9678 | 850.0506 | 423.9839 |

DBH | 0.9920 | 909.8294 | 910.0438 | 452.9147 |

Poisson | 7.8430 | 1260.3650 | 1260.4478 | 629.1825 |

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**MDPI and ACS Style**

Aljohani, H.M.; Akdoğan, Y.; Cordeiro, G.M.; Afify, A.Z.
The Uniform Poisson–Ailamujia Distribution: Actuarial Measures and Applications in Biological Science. *Symmetry* **2021**, *13*, 1258.
https://doi.org/10.3390/sym13071258

**AMA Style**

Aljohani HM, Akdoğan Y, Cordeiro GM, Afify AZ.
The Uniform Poisson–Ailamujia Distribution: Actuarial Measures and Applications in Biological Science. *Symmetry*. 2021; 13(7):1258.
https://doi.org/10.3390/sym13071258

**Chicago/Turabian Style**

Aljohani, Hassan M., Yunus Akdoğan, Gauss M. Cordeiro, and Ahmed Z. Afify.
2021. "The Uniform Poisson–Ailamujia Distribution: Actuarial Measures and Applications in Biological Science" *Symmetry* 13, no. 7: 1258.
https://doi.org/10.3390/sym13071258