Statistics 270 - Lecture 11.

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Statistics 270 - Lecture 11

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

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

Poisson Distribution Is this a pmf?

Poisson Distribution E(X) V(X)

Poisson Distribution Observations:

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

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

Poisson Approximation to the Binomial Proof:

Poisson Approximation to the Binomial Rule of thumb:

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

Example n=100 n=1000