a. In a Poisson process, what has to happen in both the time interval (0, t) and the int...

a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval (t, t + ?t) so that no events occur in the entire interval (0, t + ?t)? Use this and Assumptions 1–3 to write a relationship between P₀ (t + ?t) and P₀(t).

b. Use the result of part (a) to write an expression for the difference P₀ (t + ?t) - P₀(t). Then divide by ?t and let ?t → 0 to obtain an equation involving ( d / dt) P₀(t), the derivative P₀(t) with respect to t.

c. Verify P₀ (t) = e ^–αt that satisfies the equation of part (b).

d. It can be shown in a manner similar to parts (a) and (b) that the P_k(t)s must satisfy the system of differential equations

Verify that satisfies the system. (This is actually the only solution.)