Page 312 - Applied Probability

14. Poisson Approximation
If we let q α be the success rate per trial for category α ∈ I = {1,...,m},
then this category accumulates d or more successes with probability
n

k
=
p α
α
k
k=d n q (1 − q α ) n−k . 301
The coupling method provides a bound on the total variation distance
m

between S = W d and a Poisson random variable with mean λ = p α .
α=1
Our argument will make it clear that we can even elect a different quota d α
for each category in defining the number of categories that meet or exceed
their quotas.
To validate the coupling bound (14.1), we must construct the random
variables U α and V α described in Section 14.1. For U α we imagine conduct-
ing the multinomial trials according to the usual rules and set U α = W d .
If the number of outcomes Y α falling in category α satisfies Y α ≥ d, then

X α = 1, and we set V α = X β .If Y α <d, then we resample from
β =α
the conditional distribution of Y α given the event Y α ≥ d. This produces a
∗ ∗
random variable Y >Y α , and we redefine the outcomes of the first Y −Y α
α
α
trials falling outside category α so that they now fall in category α.Ifwe let
V α be the number of categories other than α that now exceed their quota
d, it is obvious because of the redirection of outcomes that W d ≥ V α . Thus,
the conditions for the Chen-Stein bound (14.2) apply. As pointed out in
m 2
Problem 4 of Chapter 4, the sum p should be small for the Poisson

α=1 α
approximation to have any chance of being accurate.
14.4 Construction of Somatic Cell Hybrid Panels
Somatic cell hybrids are routinely used to assign particular human genes
to particular human chromosomes [6, 21]. In brief outline, somatic cell
hybrids are constructed by fusing normal human cells with permanently
transformed rodent cells. The resulting hybrid cells retain all of the rodent
chromosomes while losing random subsets of the human chromosomes. A
few generations after cell fusion, clones can be identified with stable sub-
sets of the human chromosomes. All chromosomes, human and rodent, nor-
mally remain functional. With a broad enough collection of different hybrid
clones, it is possible to establish a correspondence between the presence or
absence of a given human gene and the presence or absence of each of the
24 distinct human chromosomes. From this pattern one can assign the gene
to a particular chromosome.
For this program of gene assignment to be successful, certain major as-
sumptions must be satisfied. First, the human gene should be present on a
single human chromosome or on a single pair of homologous human chro-
mosomes. Second, the human gene should be detectable when present in a
clone and should be distinguishable from any rodent analog of the human