For m loci in a haploid clone with no missing observations, the expected number of obligate breaks E[B(id)] is given by expression

(11.2).

(a) Under the correct order, show [1] that Var[B(id)] = 2r(1 − r)

m

−1 i=1

θi,i+1 − 2r(1 − r)

m

−1 i=1

θ2 i,i+1

+ (1 − 2r)

2 m

−2 i=1 m

−1 j=i+1

θi,i+1θj,j+1(1 − θi+1,j )

!

, where the breakage probability θi+1,j = 0 when i+1 = j. (Hint:

Let Si be the indicator of whether a break has occurred between loci i and i + 1.

Verify that E(SiSj ) = r(1 − r)θi,i+1θj,j+1[1 − θi+1,j (1 − 2r)

2]

by considering four possible cases consistent with SiSj = 1.

The first case is characterized by retention at locus i, nonretention at locus i + 1, retention at locus j, and nonretention at locus j + 1.)

(b) The above expression for Var[B(id)] can be simplified in the Poisson model by noting that 1 − θi+1,j =

j

−1 k=i+1

(1 − θk,k+1).

Using this last identity, argue by induction that m

−2 i=1 m

−1 j=i+1

θi,i+1θj,j+1(1 − θi+1,j )

= m −1 i=1 θi,i+1 − [1 −
m −1 i=1 (1 − θi,i+1)].