Index | Number | Quantity | Dependencies | Description |
---|---|---|---|---|

1 | 26 | None | Conversion of elapsed time in units of 2N_{A} generations in a growing population of size N(t) = N(0)e^{−} to elapsed time in units of 2^{αt}N_{A} generations in a diploid population of constant size N = 1 | |

2 | 27 | 1 | Probability that k lineages remain at time t (in units of N(0) generations) when n lineages are sampled at time 0 | |

3 | 4 | 2 | Joint probability that T does not coalesce by time t_{D}, and i_{D} lineages (not including T) remain at time t_{D} | |

4 | 5 | 3 | Probability that T does not coalesce before time t_{D} | |

5 | 6 | ℙ() = 1 − ℙ()^{c} | 4 | Probability that T coalesces before time t_{D} |

6 | 15 | None | Probability that T first coalesces with a lineage of type 1, given that there are i lineages of type 1, j lineages of type 2 and k lineages of type 1–2 at time t_{D} | |

7 | 16 | None | Probability that T first coalesces with a lineage of type 2, given that there are i lineages of type 1, j lineages of type 2 and k lineages of type 1–2 at time t_{D} | |

8 | 17 | None | Probability that T first coalesces with a lineage of type 1-2, given that there are i lineages of type 1, j lineages of type 2 and k lineages of type 1-2 at time t_{D} | |

9 | 1 | 6, 5, 3, 2 | Probability that T first coalesces with a lineage of type 1 | |

10 | 9 | 7, 3, 2 | Probability that T first coalesces with a lineage of type 2 | |

11 | 10 | 8, 3, 2 | Probability that T first coalesces with a lineage of type 1–2 | |

12 | 23 | 5, 2 | Survival function of the time until lineage T coalesces, given that T coalesces before time t_{D} | |

13 | 22 | 12 | Expected time until T coalesces, given that T coalesces before time t_{D} | |

14 | 24 | 4, 3 | Expected time until T coalesces with a lineage with descendants in population 1, given that T first coalesces after time t_{D} | |

15 | 21 | E[T_{1}] = E[T_{1}|]ℙ() + E[T_{1}|]ℙ(^{c})^{c} | 14, 13, 5, 4 | Expected time until T coalesces with a lineage with descendants in population 1 |

16 | 25 | 2 | Expected time until T first coalesces with a lineage with descendants in population 2 |

The derivation of each expression is the same as in the case of populations of constant size, except that

*h*_{n}_{,}(_{k}*t*;*N*(0),*α*) is used, rather than*h*_{n}_{,}(_{k}*t*;*N*). The quantities on which the expressions in the table depend are given in the Dependencies column. The numbers in the Dependencies column correspond to those in the Index column. The number of each equation—or its analog for the case of populations of constant size—is given in the Number column. Formulas for the case in which populations 1 and 2 have constant size are obtained by setting*α*_{1}and*α*_{2}equal to 0.