Difference between revisions of "Coalescent"
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The PDF of an exponential distribution of the coalescence of two lineages in a diploid population of size ''N''. | 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> | <math>P(\text{coalescence at time }g)=\frac{1}{2N}e^{-g/2N}</math> | ||
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Integrate to get the CDF. | Integrate to get the CDF. | ||
Revision as of 04:22, 5 February 2016
Coalescence of Two Lineages
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]
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 the CDF must [math]\lim_{g \to \infty}\left( -e^{-g/2N} + C \right)= 1[/math] and [math]\lim_{g \to \infty} -e^{-g/2N}= 0[/math] then [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]
