SECTION 5 5 POISSON PROBABILITY DISTRIBUTIONS POISSON DISTRUBUTION

SECTION 5. 5 POISSON PROBABILITY DISTRIBUTIONS

POISSON DISTRUBUTION Useful for calculating the probability that a certain number of events will occur over a specific period of time.

DEFINITION A POISSON DISTRIBUTION is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit.

FORMULA The probability of the event occurring x times over an interval is given by the formula

REQUIREMENTS e is a constant or the Euler's number = 2. 718 µ is the mean number of outcomes of the event over the intervals The random variable x is the number of occurrences of an event over some interval. The occurrences must be random. The occurrences must be independent. The occurrences must be uniformly distributed over the interval being used.

PARAMETERS The mean is µ The standard deviation is

PARAMETERS The Poisson Distribution is sometimes used to approximate the binomial distribution when n is large and p is small. Requirements n ≥ 100 np ≤ 10 Mean for Poisson Distribution as an approximation to the Binomial is µ = np

BINOMIAL vs POISSON A particular binomial distribution is determined by the sample size n and the probability p, but a Poisson distribution is determined only by the mean. In a binomial distribution, the possible vales of the random variable x are 0, 1, …, n but a Poisson distribution has possible x values of 0, 1, 2, … with no upper limit.

EXAMPLE Various sources provide different earthquake data, but assume that for a recent 41 year period in the US, there were 268 earthquakes measured at 6. 0 or higher on the Richter scale. Find the mean number of earthquakes per year.

EXAMPLE Various sources provide different earthquake data, but assume that for a recent 41 year period in the US, there were 268 earthquakes measured at 6. 0 or higher on the Richter scale. Find the mean number of earthquakes per year. Mean = 284 / 41 = 6. 53658537

EXAMPLE Find the probability that in a given year, there is exactly 1 earthquake in the US that measures 6. 0 or higher on the Richter scale.

EXAMPLE Find the probability that in a given year, there is exactly 1 earthquake in the US that measures 6. 0 or higher on the Richter scale. P(x=1) = 0. 009665

EXAMPLE Find the probability that in a given year, there is at least 1 earthquake in the US that measures 6. 0 or higher on the Richter scale.

EXAMPLE Find the probability that in a given year, there is at least 1 earthquake in the US that measures 6. 0 or higher on the Richter scale. P(at least 1) = 1 - P(0) P(at least 1) = 1 P(at least) = 0. 99855

EXAMPLE Is It unlikely to have a year without any earthquakes that measure 6. 0 or higher on the Richter scale? Why or why not?

EXAMPLE Is It unlikely to have a year without any earthquakes that measure 6. 0 or higher on the Richter scale? Why or why not? Yes, it is very unlikely to have a year without any earthquakes The probability of having no earthquakes is 0. 001449, which is less than 0. 05.

EXAMPLE B Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0. 000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12, 429 children. Assuming that neuroblastoma occurs as usual, find the mean number of cases in groups of 12, 429 children.

EXAMPLE B Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0. 000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12, 429 children. Assuming that neuroblastoma occurs as usual, find the mean number of cases in groups of 12, 429 children. Mean = 12, 429(0. 000011) = 0. 136719

EXAMPLE B Find the probability that the number of neuroblastoma cases in a group of 12, 429 children is 0 or 1.

EXAMPLE B Find the probability that the number of neuroblastoma cases in a group of 12, 429 children is 0 or 1. P(0 or 1) = 0. 872215 + 0. 119248 = 0. 9914636

EXAMPLE B Find the probability that the number of neuroblastoma cases in a group of 12, 429 children is more than one.

EXAMPLE B Find the probability that the number of neuroblastoma cases in a group of 12, 429 children is more than one. P(1+) = 1 – P(0 or 1) P(1+) = 1 – 0. 9914 P(1+) = 0. 0086