1. Statistics Lecture 11

2. Last day/Today: More discrete probability distributions
Assignment 4: Chapter 3: 5, 7,17, 25, 27, 31, 33, 37, 39, 41, 45, 47, 51, 65, 67, 77, 79

3. Poisson Distribution
A random variable, X, has a Poisson distribution with parameter l (l>0) if the pmf is
For x=0, 1, 2, …

4. Poisson Distribution
Is this a pmf?

5. Poisson Distribution
E(X)
V(X)

6. Poisson Distribution
Observations:

7. Example
Suppose the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter l=20
Find the probability that ten drivers are observed
Find the probability that at most 10 drivers are observed

8. Poisson Approximation to the Binomial
Suppose that the random variable X has a Bin(N,p) distribution and n!1 and p!0 so that np=l. The Bin(n,p) ! Pois(\lambda).
So, in any binomial experiment where n is large and p is small then the binomial distribution is approximately the same as the Poisson distribution where l=np
Can us Poisson distribution to compute binomial probabilities

9. Poisson Approximation to the Binomial
Proof:

10. Poisson Approximation to the Binomial
Rule of thumb:

11. Example
A rare blood disease occurs in a population with frequency 0.001
If n people are tested, what is the probability that at least two have the rare disease

12. Example
n=100
n=1000