# A Fresh Approach to a Special Type of the Luria–Delbrück Distribution

## Abstract

**:**

## 1. Introduction

## 2. The Probability Generating Function

## 3. An Integration-Based Method

## 4. A More Practical Algorithm

## 5. Asymptotic Behavior of the Mutant Probability

**Proposition 1.**

**Proposition 2.**

**Proposition 3.**

**Proposition 4.**

- 1.
- ${\sum}_{k=1}^{\infty}{f}_{i}\left(k\right)<\infty $ for $i=1,2$;
- 2.
- ${f}_{1}\left(x\right)\sim {f}_{2}\left(x\right)$ as $x\to \infty $;
- 3.
- there exists a sequence $\{{c}_{n};n\ge 1\}$ of positive constants such that ${c}_{n}\to \infty $ and $P\left({\displaystyle \bigcap _{n=1}^{\infty}}\{{Y}_{n}\ge {c}_{n}\}\right)=1.$

**Proof.**

## 6. Examples and Simulation Results

## 7. Concluding Remarks

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Distributional patterns of maximum likelihood estimates of mutation rates based on two groups of simulated experiments. Each group comprises 10,000 experiments simulated by assuming a common mutation rate of $5\times {10}^{-6}$.

**Figure 3.**The p-values generated by performing likelihood ratio tests on 10,000 pairs of simulated fluctuation experiments. Because the two mutation rates were equal, the sorted p-values exhibited an expected linear pattern. The solid line represents the observed p-values, and the dashed line represents the theoretical reference lines with slope ${10}^{-4}$ and y-intercept 0.

k | Recursive | Asymptotic | Error |
---|---|---|---|

1000 | 6.3946195 × 10${}^{-6}$ | 6.2485639 × 10${}^{-6}$ | 2.28% |

1200 | 4.6651675 × 10${}^{-6}$ | 4.5713128 × 10${}^{-6}$ | 2.01% |

1400 | 3.5743179 × 10${}^{-6}$ | 3.5097405 × 10${}^{-6}$ | 1.81% |

1600 | 2.8383575 × 10${}^{-6}$ | 2.7916453 × 10${}^{-6}$ | 1.65% |

1800 | 2.3163411 × 10${}^{-6}$ | 2.2812359 × 10${}^{-6}$ | 1.52% |

2000 | 1.9314605 × 10${}^{-6}$ | 1.9042712 × 10${}^{-6}$ | 1.41% |

2500 | 1.3147908 × 10${}^{-6}$ | 1.2989647 × 10${}^{-6}$ | 1.20% |

3000 | 9.6046479 × 10${}^{-7}$ | 9.5029417 × 10${}^{-7}$ | 1.06% |

3500 | 7.3661056 × 10${}^{-7}$ | 7.2961229 × 10${}^{-7}$ | 0.95% |

4000 | 5.8539548 × 10${}^{-7}$ | 5.8033314 × 10${}^{-7}$ | 0.86% |

4500 | 4.7803264 × 10${}^{-7}$ | 4.7422815 × 10${}^{-7}$ | 0.80% |

5000 | 3.9881054 × 10${}^{-7}$ | 3.9586392 × 10${}^{-7}$ | 0.74% |

5500 | 3.3853023 × 10${}^{-7}$ | 3.3619173 × 10${}^{-7}$ | 0.69% |

6000 | 2.9149900 × 10${}^{-7}$ | 2.8960539 × 10${}^{-7}$ | 0.65% |

7500 | 1.9865134 × 10${}^{-7}$ | 1.9754916 × 10${}^{-7}$ | 0.55% |

8000 | 1.7780098 × 10${}^{-7}$ | 1.7685851 × 10${}^{-7}$ | 0.53% |

8500 | 1.6021445 × 10${}^{-7}$ | 1.5940085 × 10${}^{-7}$ | 0.51% |

9000 | 1.4523093 × 10${}^{-7}$ | 1.4452265 × 10${}^{-7}$ | 0.49% |

9500 | 1.3235060 × 10${}^{-7}$ | 1.3172937 × 10${}^{-7}$ | 0.47% |

10,000 | 1.2118944 × 10${}^{-7}$ | 1.2064088 × 10${}^{-7}$ | 0.45% |

10,500 | 1.1144824 × 10${}^{-7}$ | 1.1096090 × 10${}^{-7}$ | 0.44% |

11,000 | 1.0289091 × 10${}^{-7}$ | 1.0245558 × 10${}^{-7}$ | 0.42% |

k | Recursive | Asymptotic | Error |
---|---|---|---|

1000 | 2.9574909 × 10${}^{-7}$ | 2.9350000 × 10${}^{-7}$ | 0.76% |

1200 | 2.0513796 × 10${}^{-7}$ | 2.0381944 × 10${}^{-7}$ | 0.64% |

1400 | 1.5058433 × 10${}^{-7}$ | 1.4974490 × 10${}^{-7}$ | 0.56% |

1600 | 1.1521612 × 10${}^{-7}$ | 1.1464844 × 10${}^{-7}$ | 0.49% |

1800 | 9.0988439 × 10${}^{-8}$ | 9.0586420 × 10${}^{-8}$ | 0.44% |

2000 | 7.3670246 × 10${}^{-8}$ | 7.3375000 × 10${}^{-8}$ | 0.40% |

2500 | 4.7113536 × 10${}^{-8}$ | 4.6960000 × 10${}^{-8}$ | 0.33% |

3000 | 3.2701091 × 10${}^{-8}$ | 3.2611111 × 10${}^{-8}$ | 0.28% |

3500 | 2.4016450 × 10${}^{-8}$ | 2.3959184 × 10${}^{-8}$ | 0.24% |

4000 | 1.8382465 × 10${}^{-8}$ | 1.8343750 × 10${}^{-8}$ | 0.21% |

4500 | 1.4521236 × 10${}^{-8}$ | 1.4493827 × 10${}^{-8}$ | 0.19% |

5000 | 1.1760123 × 10${}^{-8}$ | 1.1740000 × 10${}^{-8}$ | 0.17% |

5500 | 9.7176953 × 10${}^{-9}$ | 9.7024793 × 10${}^{-9}$ | 0.16% |

6000 | 8.1645662 × 10${}^{-9}$ | 8.1527778 × 10${}^{-9}$ | 0.14% |

7500 | 5.2239033 × 10${}^{-9}$ | 5.2177778 × 10${}^{-9}$ | 0.12% |

8000 | 4.5910062 × 10${}^{-9}$ | 4.5859375 × 10${}^{-9}$ | 0.11% |

8500 | 4.0665264 × 10${}^{-9}$ | 4.0622837 × 10${}^{-9}$ | 0.10% |

9000 | 3.6270442 × 10${}^{-9}$ | 3.6234568 × 10${}^{-9}$ | 0.10% |

9500 | 3.2551386 × 10${}^{-9}$ | 3.2520776 × 10${}^{-9}$ | 0.09% |

10,000 | 2.9376332 × 10${}^{-9}$ | 2.9350000 × 10${}^{-9}$ | 0.09% |

10,500 | 2.6644134 × 10${}^{-9}$ | 2.6621315 × 10${}^{-9}$ | 0.09% |

11,000 | 2.4276104 × 10${}^{-9}$ | 2.4256198 × 10${}^{-9}$ | 0.08% |

k | Recursive | Asymptotic | Error |
---|---|---|---|

1000 | 2.8496504 × 10${}^{-9}$ | 2.8380054 × 10${}^{-9}$ | 0.41% |

1200 | 1.8289321 × 10${}^{-9}$ | 1.8227030 × 10${}^{-9}$ | 0.34% |

1400 | 1.2571893 × 10${}^{-9}$ | 1.2535188 × 10${}^{-9}$ | 0.29% |

1600 | 9.0866594 × 10${}^{-10}$ | 9.0634442 × 10${}^{-10}$ | 0.26% |

1800 | 6.8242226 × 10${}^{-10}$ | 6.8087237 × 10${}^{-10}$ | 0.23% |

2000 | 5.2823729 × 10${}^{-10}$ | 5.2715748 × 10${}^{-10}$ | 0.20% |

2500 | 3.0711312 × 10${}^{-10}$ | 3.0661083 × 10${}^{-10}$ | 0.16% |

3000 | 1.9718894 × 10${}^{-10}$ | 1.9692016 × 10${}^{-10}$ | 0.14% |

3500 | 1.3558537 × 10${}^{-10}$ | 1.3542696 × 10${}^{-10}$ | 0.12% |

4000 | 9.8019343 × 10${}^{-11}$ | 9.7919126 × 10${}^{-11}$ | 0.10% |

4500 | 7.3626620 × 10${}^{-11}$ | 7.3559704 × 10${}^{-11}$ | 0.09% |

5000 | 5.6999368 × 10${}^{-11}$ | 5.6952742 × 10${}^{-11}$ | 0.08% |

5500 | 4.5218134 × 10${}^{-11}$ | 4.5184507 × 10${}^{-11}$ | 0.07% |

6000 | 3.6602731 × 10${}^{-11}$ | 3.6577779 × 10${}^{-11}$ | 0.07% |

7500 | 2.1286359 × 10${}^{-11}$ | 2.1274749 × 10${}^{-11}$ | 0.05% |

8000 | 1.8197713 × 10${}^{-11}$ | 1.8188408 × 10${}^{-11}$ | 0.05% |

8500 | 1.5705871 × 10${}^{-11}$ | 1.5698313 × 10${}^{-11}$ | 0.05% |

9000 | 1.3669876 × 10${}^{-11}$ | 1.3663663 × 10${}^{-11}$ | 0.05% |

9500 | 1.1987501 × 10${}^{-11}$ | 1.1982339 × 10${}^{-11}$ | 0.04% |

10,000 | 1.0583261 × 10${}^{-11}$ | 1.0578931 × 10${}^{-11}$ | 0.04% |

10,500 | 9.4005077 × 10${}^{-12}$ | 9.3968451 × 10${}^{-12}$ | 0.04% |

11,000 | 8.3961125 × 10${}^{-12}$ | 8.3929899 × 10${}^{-12}$ | 0.04% |

${\mathit{N}}_{\mathit{t}}$ | $\mathit{\u03f5}\times 100\%$ | Mutant |
---|---|---|

881,200 | 0.12 | 2 |

1,147,200 | 0.11 | 1 |

529,800 | 0.22 | 19 |

1,215,300 | 0.14 | 42 |

230,000 | 0.2 | 10 |

748,400 | 0.04 | 0 |

296,500 | 0.4 | 6 |

378,800 | 0.87 | 8 |

1,318,500 | 0.63 | 32 |

1,328,000 | 0.27 | 10 |

999,400 | 0.28 | 3 |

1,567,500 | 0.5 | 11 |

${\mathit{N}}_{\mathit{t}}$ | $\mathit{\u03f5}\times 100\%$ | Mutant |
---|---|---|

432,900 | 0.86 | 213 |

54,300 | 5.61 | 31 |

145,600 | 2.40 | 481 |

103,700 | 4.70 | 79 |

138,600 | 3.69 | 151 |

115,000 | 5.25 | 161 |

100,100 | 3.57 | 833 |

51,400 | 8.14 | 895 |

364,100 | 1.46 | 1262 |

118,800 | 3.93 | 899 |

Group | Mean of $\widehat{\mathit{\mu}}$ | Median of $\widehat{\mathit{\mu}}$ | 95% CI Coverage |
---|---|---|---|

A | 5.071 × 10${}^{-6}$ | 5.029 × 10${}^{-6}$ | 94.75% |

B | 5.010 × 10${}^{-6}$ | 5.001 × 10${}^{-6}$ | 95.30% |

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**MDPI and ACS Style**

Zheng, Q. A Fresh Approach to a Special Type of the Luria–Delbrück Distribution. *Axioms* **2022**, *11*, 730.
https://doi.org/10.3390/axioms11120730

**AMA Style**

Zheng Q. A Fresh Approach to a Special Type of the Luria–Delbrück Distribution. *Axioms*. 2022; 11(12):730.
https://doi.org/10.3390/axioms11120730

**Chicago/Turabian Style**

Zheng, Qi. 2022. "A Fresh Approach to a Special Type of the Luria–Delbrück Distribution" *Axioms* 11, no. 12: 730.
https://doi.org/10.3390/axioms11120730