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.
- Xt - Xs is Poisson.
- X(0) = 0.
- What is distribution, mean and variance of Xt?
- 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(Xt+h-Xt) = lambda(t)h
- P(Xt+h-Xt = 1) = lambda(t)h
- Example: Call board.
Dean P. Foster
Last modified: Mon Mar 22 11:51:41 EST 2004