Admistrivia

Poisson point process

Mind your names

• Poisson distribution: one random variable
• Poisson process: collection of random variables indexed by time
• Poisson point process: collection of random variables indexed by intervals

Example

• Points distributed "uniformly" on a line
• Makes more intuitive sense than mathematical sense
• N((a,b]) = number that occur AFTER a, but at or before b.

Definition

1. the number of events occuring in disjoint interval is independent.
2. (The distribution of N((t,t+h]) depends only on h and not t.)
3. P(N((t,t+h]) >= 1) = lambda h + o(h).
4. P(N((t,t+h]) >= 2) = o(h).

• Ideal notation: o_{h --> 0}(h)
• traditional notation: o_h(h)
• lazy notation: o_h(h)
• meaning: f(h) = o(h) if lim f(h)/h = 0
• Exercises:
  • o(h) + o(h) = o(h)
  • o(1) + o(h) = o(1)
  • o(h)*o(h) = o(h²)
  • etc...

KEY POINT: No mention of "Poisson" anywhere in definition

N((0,t]) is a poisson process

• Independence follows from first postulate.
• X(0) = 0 follows from properties of the empty set.
• So we need only check that X(s+t)-X(s) is Poisson
  • Chop interval up into n pieces
  • Let epsilon_i denote whether or not it contains at least one point
  • S = sum epsilon_i
  • first step: S --> Poisson as n --> infinity
  • second step: S = N((s,t]).

V.3: Distributions associated with the Poisson process

• P(t < W_k < t+h) = hf(t) + o(h)
• P(t < W_k < t+h) = P(N_t < k <= < N_t+h)
• But, we this one we know already!

Waiting times are independent and exponential