1.8. Let N(t) be a pure birth process for which Pr{an event happens in (t, t + h)IN(t) is odd} = ah + o(h), Pr{an event happens in (t, t + h)IN(t) is even} _ 6h + o(h), where o(h)/h -* 0 as h 1 0. Take N(O) = 0. Find the following probabilities:

Po(t) = Pr{N(t) is even}; P,(t) = Pr(N(t) is odd}.

Hint: Derive the differential equations Po(t) = aP,(t) - /3Po(t) and P,(t) = -aP,(t) + /3Po(t), and solve them by using Po(t) + P,(t) = 1.