2.20 A study of the mortality of grey herons involves the release of a cohort of birds marked soon after hatching in a single year. The probabilities of birds being reported dead in each of the following four years are partially shown below in temporal order, for a model with a constant reporting probability λ, a probability φ1 that any bird survives its first year of life and a probability φa that any bird survives any later year of life

(1 − φ1)λ φ1(1 − φa) ∗ φ1φa(1 − ∗) ∗ φ1φa ∗ (1 − φa)λ.

(i) Complete the last three expressions, by replacing the ∗’s by the appropriate probabilities. Show that the probability of a bird not being found during these four years is given by 1 − λ(1 − φ1φ3 a).

(ii) The data below relate to a cohort of grey herons marked in Britain in 1957, and form a small subset of the data in Table 1.5.