Problem 1.3. As a complication of the Rutherford–Chadwick–Ellis experiment, suppose that the number of particles that decay in each interval is registered by a counter. We assume that events occur according to a Poisson process with intensity λ > 0, but whenever the counter registers an event, it is inoperative for the next b ≥ 0 units of time (and does not register any new events in that time interval). Let R(t) denote the number of events that are registered by time t ≥ 0.
(i) What is the probability that the first k events are all registered, for each k = 1, 2, . . .?
(ii) For t ≥ (n − 1)b, find P{R(t) ≥ n}.
Many thanks! first part of my solution
After an event is registered, the probability that no event occurs during the latency period is $\mathrm e^{-\lambda b}$. The probability for this to happen $k$ times in a row is $\mathrm e^{-k\lambda b}$.
For $t\ge(n-1)b$, there is enough time for $n-1$ latency periods, and they take up a time $(n-1)b$, leaving $t-(n-1)b$. The probability for at least $n$ events to occur in this time is
$$ \mathrm e^{-\mu}\sum_{k=n}^\infty\frac{\mu^k}{k!}=1-\mathrm e^{-\mu}\sum_{k=0}^{n-1}\frac{\mu^k}{k!} $$
with $\mu=\lambda(t-(n-1)b)$.