10.9 A device recording lightning intensity is attached to a pole during a thunderstorm. Assume that during that time the bolts of lightning occur at that pole according to a Poisson process with rate λ. However, each bolt registered by the machine renders the device inoperative for a fixed length of time a and it does not register anything that may occur during that interval. Let R(t) denote the number of lightning bolts that are registered by the machine at time t.

a) Find the probability that the first k bolts of lightning are all registered.

b) For t ≥ (n − 1)a, find P{R(t) ≥ n}

Hint: Look to Type I Type II events for the Poisson process.