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

Chapter 3, Problem 93

(choose chapter or problem)

a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval \((t,t+ \Delta t)\) so that no events occur in the entire interval \((0,t+ \Delta t)\)?  Use this and Assumptions 1–3 to write a relationship between \(P_0(t+ \Delta t)\) and \(P_0(t)\).

b. Use the result of part (a) to write an expression for the difference \(P_0(t+ \Delta t)-P_0(t)\). Then divide \(\Delta t\) and let \(\Delta t \longrightarrow 0\) to obtain an equation involving \((d/dt)P_0(t)\), the derivative of \(P_0(t)\) with respect to t.

c. Verify that \(P_0(t)=e^{-\alpha t}\) satisfies the equation of part (b).

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

\(\begin{aligned} \frac{d}{d t} P_{k}(t) &=\alpha P_{k-1}(t)-\alpha P_{k}(t) \\ k &=1,2,3, \ldots \end{aligned} \)

Verify that \(P_k(t) = e^{-\alpha t} (\alpha t)k /k!\) satisfies the system. (This is actually the only solution.)