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

b. Use the result of part

(a) to write an expression for the difference . Then divide by t and let to obtain an equation involving , the derivative of P0(t) with respect to t.

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

Pk(t) 5 e2at

(at)k

/k!

k 5 1, 2, 3, . . .

d dt Pk(t) 5 aPk21(t