The expected time to coalesce, in generations, is the same as the ploidy (e.g., 2 for humans) times Nef, the inbreeding effective population size, under coalescent theory. Why?

Both ploidy * Nef and the expected time to coalesce are equal to the summation from generation $t=1$ to infinity of (t(1-((1/(ploidy * Nef)^(t-1) * (1/(ploidy * Nef)))))), or written more prettily:

$$\sum_{t=1}^{\infty} t\left(1-\left(\frac{1}{\text{ploidy} \cdot N_{ef}}\right)^{t-1} \cdot \left(\frac{1}{\text{ploidy} \cdot N_{ef}}\right)\right).$$

Why, and what does it mean to have an "infinite" generation (in the summation)?

The time to the last coalescent (of two gene copies) is ploidy * Nef.

If you multiply this whole summation by the number of pairs of genes, the result is a new summation which represents the sum over generations of the probability that coalescence occurs in the t-th generation, or the probability of the first coalescence occuring within t generations.



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