6.1.8 Let N.t/ be a pure birth process for which Prfan event happens in.t; tCh/jN.t/ is oddg D hCo.h/;

Prfan event happens in.t; tCh/jN.t/ is eveng D hCo.h/;

where o.h/=h!0 as h # 0. Take N.0/ D 0. Find the following probabilities:

P0.t/ D PrfN.t/ is evengI P1.t/ D PrfN.t/ is oddg:

Hint: Derive the differential equations P0 0.t/ D P1.t/????P0.t/ and P0 1.t/ D ????P1.t/CP0.t/

and solve them by using P0.t/CP1.t/ D 1.