Page 286 - Applied Probability

12. Models of Recombination
274
and mixture models have skip distributions determined by s 5 = 1 and
(s 4 ,s 5 )= (.06,.94), respectively. These two models yield a further very
large improvement over the count-location model because they take into
account position interference as well as count interference. In spite of the
inadequacies of Haldane’s model and the count-location model, all four
models give roughly similar estimates of the map distances (in centiMor-
gans = 100 × Morgans) between adjacent pairs of loci. Note that Haldane’s
model and the count-location model compensate for the reduced number
of double crossovers on adjacent intervals by expanding map distances.
TABLE 12.2. Analysis of the Morgan et al. Data
Interval
Model 1 2 3 4 5 6 7 8 max ln L
Haldane 5.4 11.4 8.5 15.2 8.9 18.4 8.1 4.6 −37956.61
Count-Loc 5.3 10.8 8.2 14.2 8.6 16.9 7.8 4.5 −37449.17
Chi-square 5.1 9.8 7.5 13.3 8.4 15.6 7.5 4.4 −36986.87
Mixture 5.1 9.8 7.5 13.3 8.4 15.5 7.5 4.4 −36986.34

12.8 Problems

1. Prove that in Haldane’s model the gamete probability formula (12.6)
collapses to the obvious independence formula

k

s i
= θ (1 − θ i ) 1−s i .
y S
i
i=1
2. Karlin’s binomial count-location model [12] presupposes that the to-
tal number of chiasmata N has binomial distribution with generating
1
s r
function Q(s)= ( + ) . Compute the corresponding map function
2 2
and its inverse.
3. Consider a delayed renewal process generated by the sequence of inde-
pendent random variables X 1 ,X 2 ,... such that X 1 has distribution
function G(x) and X i has distribution function F(x) for i ≥ 2. If
G(x)= F(x), then show that the renewal function U(x)=E(N [0,x] )
satisfies the renewal equation
x

U(x)= F(x)+ U(x − y)dF(y). (12.20)
0