Suppose that {N(t),t ≥ 0} is a Poisson process with rate λ > 0. For a fixed time instant t > 0, we consider the random variables A(t) and D(t)
(a) Calculate the distribution of both A(t) and D(t).
(b) What is the distribution of A(t) + D(t)?
(c) Suppose now that we consider the Poisson process over the entire real line, that is, {N(t),t ∈ R}. What then is the distribution of A(t) + D(t)?
(d) In (c), we can interpret the sum A(t) + D(t) as being the length of the interval, between two events, which contains the fixed time instant t. Explain why the distribution of this sum is not an exponential distribution with parameter λ.