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12. Models of Recombination
9. In the count-location model, suppose that the count distribution is
concentrated on the set {0, 1} or on the set {1, 2, 3, 4}. Prove that
the model exhibits positive or no interference [23]. (Hint: Check the
2
log-concavity property by showing that Q (s)Q(s) − Q (s) reduces

to a sum of negative terms.)
10. In the count-location model, suppose that the count distribution has
generating function
8 1 4 4 5
Q(s)= s + s + s .
35 5 7
Prove that the opposite of inequality (12.16) holds for x 1 = .4 and
x 2 = .1 and, therefore, that negative interference occurs [23].
11. The Poisson-skip model with skip distribution s 1 = p and s 3 =1 − p
has interarrival density
x 2 −x
−x
f(x) = pe +(1 − p) e .
2
Show that f(x) has decreasing hazard rate for x small and positive
[18].
12. The collection of count-location models that are also stationary re-
newal models is very small [3]. Haldane’s homogeneous Poisson model
is one example. Another is the trivial model with no chiasmata. The
admixture of two such independent processes furnishes a third exam-
ple subsuming the first two. To prove that this exhausts the possi-
bilities, consider a count-location process that uniformly distributes
its pointson[0, 1]. Let q n denote the nth count probability. If the
process is also a stationary renewal process, let F ∞ (x) be the distri-
bution of X 1 and F(x) be the distribution of the subsequent X i ,in
the notation of Section 12.4. Show that
∞

F ∞ (x) = 1 − q n (1 − x) n
n=0
and that
F(x) = Pr(X 2 ≤ x | X 1 = y)
∞ nq n (1 − x − y)
n−1
n=0
=1 − .
∞ nq n (1 − y) n−1
n=0
Use these identities and the identity F (x)= [1−F(x)]/µ to demon-

∞

strate that φ(x)= ∞ nq n (1 − x) n−1 satisfies the functional equa-
n=0
tion φ(x)φ(y)= µ −1 φ(x + y), where µ is the mean of F(x). Setting
ψ(x)= µÐ (x), it follows that ψ(x)ψ(y)= ψ(x + y) and ψ(0)=1.