The Poisson Distribution We can use the Poisson distribution to estimate the probability of arrivals at a car wash in one hour or the number of leaks in 100 miles of pipeline. Bell Labs uses it to model the arrival of phone calls.

The Poisson Distribution The Poisson distribution is defined by: Where f(x) is the probability of x occurrences in an interval  is the expected value or mean value of occurrences within an interval e is the natural logarithm. e =

Properties of the Poisson Distribution 1.The probability of occurrences is the same for any two intervals of equal length. 2.The occurrence or nonoccurrence of an event in one interval is independent of an occurrence on nonoccurrence of an event in any other interval

Example: Mercy Hospital Poisson Probability Function Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? MERCY

Example: Mercy Hospital n Poisson Probability Function  = 6/hour = 3/half-hour, x = 4

Using Excel to Compute Poisson Probabilities n Formula Worksheet … and so on

n Value Worksheet Using Excel to Compute Poisson Probabilities … and so on

n Poisson Distribution of Arrivals Example: Mercy Hospital Poisson Probabilities Number of Arrivals in 30 Minutes Probability actually, the sequence continues: 11, 12, …

Problem 31, p. 229 a.Write the appropriate Poisson distribution b.What is the average number of occurrences in three time periods? c.Write the appropriate Poisson function to determine the probability of x occurrences in three time periods. d.Compute the probability of two occurrences in one time period. e.Compute the probability of six occurrences in three time periods. f.Compute the probability of five occurrences in two time periods. Consider a Poisson probability distribution with an average number of occurrences of two per period.

Problem 31, p. 229 (a) (b) (c) (d)

Problem 31, p. 229 (e) (d)

Problem 31, p. 229

The Hypergeometric Distribution This is similar to the binominal distribution except: (1) the trials are NOT independent; and (2) the probability of success (ρ) changes from trial to trial.

Hypergeometric Distribution Let r denote in the population size N labeled a success. N – r is the number of elements in the population labeled failure. The hypergeometric distribution is used to compute the probability that in a random selection of n elements, selected without replacement, we obtain x elements labeled success and N – x elements labeled failure.

Notice that the x successes must be pulled from the r number of successes in the population and the n - x failures must be drawn from a population of N – r failures

Hypergeometric Distribution Where n = the number of trials. N = number of elements in the population r = number of elements in the population labeled a success

Hypergeometric Distribution Number of ways a sample of size n can be selected from a population of size N Number of ways a sample of size x successes can be selected from a population of size r Number of ways a sample of size n -x failures can be selected from a population of size N -r

Example: Neveready Hypergeometric Probability Distribution Bob Neveready has removed two dead batteries from a flashlight and inadvertently mingled them with the two good batteries he intended as replacements. The four batteries look identical. Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries? ZAP ZAP ZAP ZAP

Example: Neveready Hypergeometric Probability Distribution where: x = 2 = number of good batteries selected x = 2 = number of good batteries selected n = 2 = number of batteries selected n = 2 = number of batteries selected N = 4 = number of batteries in total N = 4 = number of batteries in total r = 2 = number of good batteries in total r = 2 = number of good batteries in total

Using Excel to Compute Hypergeometric Probabilities n Formula Worksheet

n Value Worksheet Using Excel to Compute Hypergeometric Probabilities