There are many variations on the preceding arguments. One can consider the same ideas with ordered or unordered tetrads, or with half-tetrads; one can discuss more than two loci, including loci on different arms of the same chromosome and on different chromosomes; one can replace the Poisson model by a more general point process model, such as a renewal process, and chi-square renewal processes have proved useful in this context; one can generalize NCI to permit the non-sister chromatid pair chosen for any exchange to depend on the chromatids chosen in adjacent exchanges; and one can include sister chromatid exchange. We will explore some of these ideas in exercises, and mention others next week.
Exercise 6. Suppose that diploid cells from a fungus having ordered tetrads is
undergoing meiosis, and that we are interested in the proportion of second
division segregants (SDS). Let N be the number of exchanges on the four strand
bundle between the locus and its centromere. Under NCI, prove that
the probability that a tetrad has second division segregation, given N=n, is
Deduce that under the Poisson model introduced earlier,
the unconditional probability of SDS is
where 2d is the expected number of exchanges between the locus and its
centromere.
(Hint: As with an earlier exercise, seek the number 
of the 
strand configurations which lead to SDS, obtaining the recursion
where 
is the
number leading to first division segregation (FDS). An inductive proof
of these relations will use a knowledge of whether the nth and 
th
exchanges are on 2, 3 or 4 strands.
Exercise 7. Formulate and prove an analogue of Mather's formula for 3 loci along a chromosome arm. Generalize to L loci.