Now we can calculate the covariance between half-siblings. To do so
we imagine selecting huge number of half-sibs pairs at random. Now
the phenotype of the first half-sib in the pair is a random variable
(call it
), as is the second (call it
). The mean of
is just the mean phenotype in all the progeny taken together,
. Similarly, the mean of
is just
. Now with
one locus, two alleles we have three possible phenotypes:
(corresponding to the genotype
),
(corresponding to
the genotype
), and
(corresponding to the genotype
). So all we need to do to calculate the covariance between
half-sibs is to write down all possible pairs of phenotypes and the
frequency with which they will occur in our sample based on the table
above and the frequency of maternal genotypes. It's actually a bit
easier to keep track of it all if we write down the frequency of each
maternal genotype and the frequency with which each possible
phenotypic combination will occur in her progeny.
![\begin{eqnarray*}
\mbox{Cov}(S_1,S_2) &=& p^2[p^2(x_{11} - {\bar x})^2 + 2pq(x_...
...2\alpha_2 - {\bar x}]^2\} \\
&=& \left({1 \over 4}\right)V_a
\end{eqnarray*}](img26.png)