In this problem, our goal is to complete the proof of the equivalence of the first and
Question:
In this problem, our goal is to complete the proof of the equivalence of the first and the second definitions of the Poisson process. More specifically, suppose that the counting process {N(t), t ∈ [0,∞)} satisfies all the following conditions:
1. N(0) = 0.
2. N(t) has independent and stationary increments.
3. We have
We would like to show that N(t) ∼ Poisson(λt). To this, for any k ∈ {0, 1, 2,⋯}, define the function gk(t) = P(N(t) = k).
a. Show that for any Δ > 0, we have
b. Using Part (a), show that
c. By solving the above differential equation and using the fact that g0(0) = 1, conclude that
d. For k ≥ 1, show that
e. Using the previous part show that
which is equivalent to
f. Check that the function
satisfies the above differential equation for any k ≥ 1. In fact, this is the only solution that satisfies g0(t) = e−λt, and gk(0) = 0 for k ≥ 1.
Step by Step Answer:
Introduction To Probability Statistics And Random Processes
ISBN: 9780990637202
1st Edition
Authors: Hossein Pishro-Nik