a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval (t, t + Δ t) so that no events occur in the entire interval (0, t + Δt)? Use this and Assumptions 1-3 to write a relationship between P0(t + Δt) and P0(t).
b. Use the result of part (a) to write an expression for the difference P0(t + Δt) - P0(t). Then divide by Δt and let Δt †’ 0 to obtain an equation involving (d/dt)P0(t), the derivative of P0(t) with respect to t.
c. Verify that P0(t) = e-at satisfies the equation of part (b).
d. It can be shown in a manner similar to parts (a) and (b) that the Pk(t)s must satisfy the system of differential equations

Verify that Pk(t) = e-at(at)k/k! satisfies the system.

dt k 1,2,3,...