# Theoretical distributions

```{tabbed} Time to most recent common ancestor
<iframe width="100%" height="476" frameborder="0"
  src="https://observablehq.com/embed/@molpopgen/kingman?cells=plot_ftmrca_data"></iframe>

The distribution of the time to the most recent common ancestor, in units of $2N_e$ generations.
For a sample of $n$ haploid lineages, the distribution is $n - 2$ convolutions of exponential random variables.
See {cite:t}`Wakeley2008-hd`, page 77, for details.
```

```{tabbed} Total time on the tree
<iframe width="100%" height="476" frameborder="0"
  src="https://observablehq.com/embed/@molpopgen/kingman?cells=plot_fttot_data"></iframe>

The distribution of the total time on a non-recombining tree, in units of $2N_e$ generations.
The plot is based on Equation 3.34 from {cite:t}`Wakeley2008-hd`, page 78.
```

```{tabbed} Number of segregating sites
<iframe width="100%" height="700" frameborder="0"
  src="https://observablehq.com/embed/@molpopgen/kingman?cells=plot_P_s_equals_k_2d%2Cviewof+max_k%2Cviewof+n_min%2Cviewof+n_max%2Cviewof+theta"></iframe>

The distribution of the number of polymorphic mutations ("segregating sites") on a tree under the infinitely-many sites model.
The formula plotted is from {cite:t}`Tavare1984-fm`, which can be found on page 94 of {cite:t}`Wakeley2008-hd`.
```