Question 1: Poisson Process Puzzles

Consider a Poisson process with rate parameter λ . This is the standard greek letter to use — in lectures it was called θ just to remind you that it’s the parameter. In this question, you will look at two different ways of formulating the data proposition. Throughout the question, use a log-uniform improper prior p(λ) ∝ 1/λ .

Suppose that, in the interval from t = 0 to t = ∆t, the number of events is x. The sampling distribution is:

x|λ ∼ Poisson(λ∆t).                                                      (1)

Here’s my picture of a Poisson process from lectures:

(a) Find the posterior distribution for λ given x, and show that it is a Gamma distribution.

(b) One property of Poisson processes is that the inter-event times have independent expo-

nential distributions with rate λ . Let the event times be times be t1 ,t2 ,t3 , and so on, so the inter-event times are d1  = (t1 − 0),d2  = (t2 − t1 ), et cetera. What is the probability distribution of tx  (the occurrence time of the xth event) given λ? I just want you to write it down (or look up the result), not derive it.

(c) Use the result from (b) to write down the joint density p(tx ,tx+1 |λ).

(d) An alternative form of presenting the data is to note that the xth event occurred before the end of the time interval [0, ∆t], AND event x + 1 must occur after the end of the interval. Use the result from (c) to find P(tx  < ∆t,tx+1  > ∆t|λ).