Admistrivia

• Posted homework 8 on the web.
• Read p 267 - 274.

Poisson distributions

Infinitely divisable distributions

• Suppose I can find A and B such that:
  • X = A + B.
  • A, B are independent.
  • A, B have the same distribution.
  Called "divisable."
• Suppose I can repeat this process as much as I like. THen it is called infinitely divisable.
• This allows construction of a continuous time Markov chain.

The poisson distribution: Basic facts

• review: density, mean and variance.
• X= Poisson(mu), Y= Poisson(lambda), X+Y = Poisson(mu+lambda)
• N poisson. M|N = Binomial(p,N). Then M is Poisson.

The Poisson process

• Definition:
  • independent increments.
  • X_t - X_s is Poisson.
  • X(0) = 0.
• What is distribution, mean and variance of X_t?
• Relies on infinite diviability.

Examples

• Defects along DNA. (SNIPS)
• Sime times it works, other times not:
  • Good model for bikers in triathelone.
  • Bad model for bikers in tour de france.
  • Why?

Nonhomogeneous process

• lambda(t) instead of lambda is rate
• E(X_t+h-X_t) = lambda(t)h
• P(X_t+h-X_t = 1) = lambda(t)h
• Example: Call board.