Transcript 5-4
5-4 Geometric and Poisson Distribution Geometric Distribution We’ve talked about these already. In a geometric distribution the goal is looking for first success on a particular trial (rather than the number of successes on a particular trial run) Geometric Distribution We’ve talked about these already. In a geometric distribution the goal is looking for first success on a particular trial (rather than the number of successes on a particular trial run) P(X x) q μ 1 p and σ p(1-p)x-1 x 1 p q p 2 1 p p Poisson Distribution This type of distribution is slightly different. In a Poisson Distribution the probability of success gets smaller and smaller as the number of trials gets larger and larger. Again, there are two outcomes (success or failure) and the events must be independent. These are events that happen over time, volume, area, etc… (like arrival times of babies, airplanes, etc) That is, waiting times. Poisson Distribution If λ (lambda) is the mean number of successes over whatever interval and r is the number of successes being estimated, then Poisson Distribution If λ (lambda) is the mean number of successes over whatever interval and r is the number of successes being estimated, then P(r) e λ λ r! r Poisson Distribution If λ (lambda) is the mean number of successes over whatever interval and r is the number of successes being estimated, then With P(r) e m ean=λ σ λ λ λ r! r Calculator Distr: poissonpdf(λ,r) = the probability of r successes Distr: poissoncdf(λ,r) = the probability at least r successes How to tell a Poisson Distribution In our homework it will say “Explain why a Poisson Distribution would be a good choice for the probability distribution of r” Otherwise, this will be a little seen topic. Notes This chapter is based on Success/Failure situations ONLY These have all been Binomial Distributions The Poisson Distribution can be a very good approximation for a Probability Distribution, provided n ≥ 100 and np < 10. Example At Burnt mesa Pueblo, in one of the archaeological excavation sites, the artifact density was 1.5 (artifacts per 10 liters of sediment). Suppose you are going to dig up and examine 50 liters of sediment. Let r = 0, 1, 2, 3… be a random variable that represents the number of prehistoric artifacts found in your 50 liters of sediment. A) What is λ? Write out the formula for the probability distribution of r. B) Compute the probability that in your 50 liters of sediment you will find 2 artifacts; 3 artifacts; 4 artifacts C) Find the probability that you will find 3 or more artifacts in the 40 liters. D) Find the probability that you will fewer than 3 artifacts.