2. Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley Follows Jim Pitman’s book: Probability Section 3.5

3. The Poisson (  ) Distribution The Poisson distribution with parameter  or Poisson(  ) distribution is the distribution of probabilities P  (k) over {0,1,2,…} defined by:

4. Poisson Distribution Recall that Poisson(  ) is a good approximation for N = Bin(n,p) where n is large p is small and  = np.

5. Poisson Distribution Example: N galaxies : the number of super- nova found in a square inch of the sky is distributed according to the Poisson distribution.

6. Poisson Distribution Example: N drops : the number of rain drops falling on a given square inch of the roof in a given period of time has a Poisson distribution. X Y 0 1 01

7. Poisson Distribution:  and  Since N = Poisson(  ) » Bin(n,  /n) we expect that: Next we will verify it.

8. Mean of the Poisson Distribution By direct computation:

9. SD of the Poisson Distribution

10. Consider two independent r.v.’s : N 1 » Poisson (  1 ) and N 2 » Poisson (  2 ). We approximately have: N 1 » Bin(  1 /p,p) and N 2 » Bin(  2 /p,p), where p is small. So if we pretend that  1 /p,  2 /p are integers then: N 1 + N 2 ~ Bin((  1 +  2 )/p,p) ~ Poisson(  1 +  2 ) Sum of independent Poissions

11. Claim: if N i » Poisson (  i ) are independent then N 1 + N 2 + … + N r ~ Poisson(  1 + …  r ) Proof (for r=2): Sum of independent Poissions

12. Poisson Scatter The Poisson Scatter Theorem: Considers particles hitting the square where: Different particles hit different points and If we partition the square into n equi-area squares then each square is hit with the same probability p n independently of the hits of all other squares

13. Poisson Scatter The Poisson Scatter Theorem: states that under these two assumptions there exists a number > 0 s.t: For each set B, the number N(B) of hits in B satisfies N(B) » Poisson( £ Area(B)) For disjoint sets B 1,…,B r, the numbers of hits N(B 1 ), …,N(B r ) are independent. The process defined by the theorem is called Poisson scatter with intensity

14. Poisson Scatter

17. Poisson Scatter Thinning Claim: Suppose that in a Poisson scatter with intensity each point is kept with prob. p and erased with probability 1-p independently of its position and the erasure of all other points. Then the scatter of points kept is a Poisson scatter with intensity p.