Admistrivia

Models

Examples

• Some times it works, other times not:
  • Good model for bikers in triathelone.
  • Bad model for bikers in tour de france.
  • Why?
• BUT: Triathelon's are a 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.

Theorem V.2.1 (p 280): Coupling argument

• epsilon_i is bernoulli: P(epsilon_i = 1) = p_i.
• epsilon_i are independent.
• S = sum epsilon_i

Proof

• NOTE: If all p's are the same--this is the traditional poisson approximation. Just do it!
• Alternative proof: if n = 1, and p is small, Binomial = bernulli = Poisson. Now sum the Poissons to discover that S is approximately Poisson. But how accurate is this?
  • Each is off by p²
  • total error is off by np² = lambda p.
• What is going on in second proof?
  • Sequence of epsilons
  • Sequence of X's that approximate the epsilons
  • They are "coupled" to live on the same probability space. (New trick that we didn't know when I was a kid.)
  • sum X's = sum epsilon's unless at least one differ.
  • bound probability of at least one differing by sum