3.4 Consider the microbial infection data of Example 1.2. A possible model for the progress of the infection is that a single infecting bacterium living according to a stochastic linear birth-and-death process (see Cox and Miller, 1965, p.156) with respective birth and death rates λ and μ, with λ>μ, produces the symptoms of a sore throat if and only if the resulting colony grows to size n. Then the time, T , to the onset of symptoms is governed by:

pr(T ≤ t) ≈ (1 − α)βn−1(1 − μ

λ)

−1

(Morgan and Watts, 1980), where

α = μ{1 − e(μ−λ)t

}{λ − μe(μ−λ)t

}−1,

β = λ{1 − e(μ−λ)t

}{λ − μe(μ−λ)t

}−1.

Verify that the likelihood is given by:

L(λ, μ, n) = 

13 i=1

{pr(T ≤ i) − pr(T ≤ i − 1)}ri , where ri is the number of individuals in the ith class interval, and try to obtain the maximum-likelihood estimates of λ, μ and n.