## Derivation of the inbreeding effective population size N_{e}

The **inbreeding
coefficient** **F **in any closed, finite population
will increase over time, as an increasing proportion of
individuals will necessarily come to have related parents. The
**effective population size** *N*_{e}
is defined by observing how the *observed ***inbreeding coefficient**
changes from one generation to the next, and defining *N*_{e}
as the size of the *ideal* population with the same
change as the *real *population under
consideration.

In any
finite population, the inbreeding coefficient **F** in the
current generation** t** changes from the previous
generation *t-1* as a **recursion equation**:

- $F_{t}={\frac {1}{N}}\left({\frac
{1+F_{t-2}}{2}}\right)+\left(1-{\frac
{1}{N}}\right)F_{t-1}.$

The equation
has two terms, (**1/N**) and its complement (**1 - 1/N**).

(1) For a **draw
& replace experiment** in a population size **N**
in the *previous *generation **t-1**, for
any individual drawn at random, the expectation of drawing the
*same **individual a second time *is **1****/N**,
and if so the expectation of drawing the *same allele*
is **1/2**. Therefore, the expectation of drawing the *same
*allele *twice *so as to obtain an individual with
two alleles that are **identical by
descent** (**I by D**)
is **1/2N**. This calculation is weighted by the
expectation **F**_{t-2} that
any pair of alleles were *already ***I by D** in the
*next previous *generation.

(2) The
expectation of *not *drawing the same individual *twice
*is **(1 - 1/N)**, weighted by the expectation **F**_{t-1}
that any random pair of alleles is *already*** ****I
by D** in the *previous *generation.

To extend
this recursion calculation from *two *generations to**
t** generations, consider the complement of the **inbreeding
coeff****icient ***F* as the **Panmixia Index** *P* = **(1 − ***F***)**,
which is the expectation that any two individuals in a finite
population size **N **do *not *share a common
ancestor. For an initial **P**_{0},
the expectation that any two alleles drawn in the subsequent
generation are *not ***I by D** is **(1 - 1/2N)**.
The same is true in each subsequent generation, so that if
population size **N **is constant, the *joint
probability* that this is *never *so over all
generations **i = 1, 2, 3, ..., t** is simply the
product of all those "*not*" terms,

Text material © 2020 by Steven M. Carr