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=The Coalescence of Two Lineages=

The chance two lineages coalesce in the previous generation is the chance that they pick the exact same copy of a gene to inherit which is <math>1/2N</math> because there are 2''N'' total copies of a gene in the population. 

The chance they did not coalesce is therefore the remaining probability <math>1-1/2N</math>. 

Each ''g'' generation there is the same independent probability of not coalescing, a total of <math>(1-1/2N)^g</math>, until a coalescent event occurs with probability <math>1/2N</math>.

Therefore, the probability of coalescence in the ''g'' generation is 

<math>P(\text{coalescence at time }g)=\frac{1}{2N}\left( 1 - \frac{1}{2N} \right)^g</math>

which is a form of geometric distribution in discrete time. 

In a large population over many generations this is closely approximated by an exponential distribution 

<math>P(\text{coalescence at time }g)=\frac{1}{2N}\left( 1 - \frac{1}{2N} \right)^g \approx \frac{1}{2N}e^{-g/2N}</math>



The PDF of an exponential distribution of the coalescence of two lineages in a diploid population of size ''N''.

<math>P(\text{coalescence at time }g)=\frac{1}{2N}e^{-g/2N}</math>

For example, the probability of two lineages coalescing in a small population of 20 individuals in exactly the ninth generation is 2%.


Integrate to get the CDF. 

<math>F(\text{coalescence at time }g)=\int_0^g\frac{1}{2N}e^{-g/2N}</math>

<math>F(\text{coalescence at time }g)=\frac{1}{2N}\int_0^g e^{-g\frac{1}{2N}}</math>

<math>F(\text{coalescence at time }g)=\frac{1}{2N} \frac{-e^{-g\frac{1}{2N}}}{\frac{1}{2N}} + C</math>

<math>F(\text{coalescence at time }g)=-e^{-g/2N} + C</math>

Because by definition the CDF area must sum to one, <math>\lim_{g \to \infty}\left( -e^{-g/2N} + C \right)= 1</math>, and the limit of <math>\lim_{g \to \infty} -e^{-g/2N}= 0</math> then the constant of integration must be one, <math>C = 1</math>.

<math>F(\text{coalescence at time }g)=-e^{-g/2N} + 1</math>

<math>F(\text{coalescence at time }g)=1-e^{-g/2N}</math>

For example, there is a 95% probability that two lineages will coalesce within 6''N'' generations. 

<math>F(\text{coalescence at time }g)=0.95=1-e^{-g/2N}</math>

<math>1-0.95=e^{-g/2N}</math>

<math>\log_e 0.05=-\frac{g}{2N}</math>

<math>-2N\log_e 0.05=g</math>

<math>-2N\times-3=g</math>

<math>6N=g</math>