107.

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

so that no events occur in the entire interval (0, tt)? Use this and Assumptions 1—3 to write a relationship between P0(tt) 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 tS0 to obtain an equation involving (d/dt)P0(t), the derivative of P0(t)

with respect to t.

c. Verify that P0(t)  eat satis es 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 d

dt Pk1t 2  aPk11t 2  aPk1t 2 k  1, 2, 3, . . .

Verify that Pk(t)  eat(at)k/k! satis es the system.
(This is actually the only solution.)