Admistrivia

• Read 297-305
• Read V.3 on your own.
  • Gamma distribution for waiting times
  • exponential distribution for inter-arrival times
  • Binomial distribution for Xs|Xt

Relationship to the uniform distribution

Approximation

• Pick up n darts and throw them at interval of length t
• Look at picture. Looks like a Poisson process.
• Each spot is equally likely (work out P(N(t,t+h)=1))
• no spot has two darts (work out P(N(t,t+h)>1))
• But not independent.
  • Divide into intervals
  • work out probably
  • Ends up with slight negative dependency

• If you pick up a poisson number of darts then counts are independent!
• P(X,Y) = P(X,Y|N)P(N) ends up with independent Poissons
• We don't "care" that they are Poissons--the independence is enough to show we have a poisson point process

Theorem

THEOREM: f(w1,w2,...,wn|N(t)=n) = n!t^-n.

Proof:

• We can generate a Poisson process any way we like and it is the same, so lets use our "poissonization" of a uniform distribution.
• U₁ is where first dart landed, etc
• W₁ is where left most dart, etc
• Given N(t)=n, the U's are uniform.
• But, the W's are NOT uniform! (P(W₁ < t/2) > 1/2)
• But all we lose is the order--hence the n!.

Example 1: NPV of Poisson payments

By the book

Example 2: Shot noise

Obviously, same if you use a Poisson number of U's. But that is easier to work with.

Time permitting: do details

• Gamma distribution for waiting times
• exponential distribution for inter-arrival times
• Binomial distribution for Xs|Xt