AP Statistics Friday, 04 December 2015 OBJECTIVE TSW (1) explore Poisson distributions, and (2) quiz over discrete distributions and binomial distributions. ASSIGNMENTS DUE –WS Binomial Distributions #2  wire basket –WS Binomial Activity  to the right of the black tray ASSIGNMENTS DUE MONDAY –WS Geometric Distributions –WS Geometric Activity –WS Poisson Distributions

Special Discrete Distributions: Poisson Distributions

This distribution deals with the probabilities of rare events that occur infrequently in space, time, distance, area, etc. Examples: The number of accidents that occur per month at a given intersection. The number of tardies per semester for a given student. The number of runs per inning in a baseball game.

Properties of Poisson Distributions: The occurrence of a success in any interval is independent of that in any other interval The probability that a success will occur in any interval is the same for all intervals of equal size and is proportional to the size of the interval We observe a discrete number of events in a continuous (fixed) interval.

Formulas for Poisson: X = number of rare events per unit of time, space, etc.  = mean value of X (Greek letter lambda)

The number of accidents in an office building during a four-week period averages 2. What is the probability there will be one accident in the next four-week period? What is the probability that there will be more than two accidents in the next four-week period?

The number of calls to a police department between 8 pm and 8:30 pm on Friday averages 3.5. What is the probability of no calls during this period? What is the probability of no calls between 8 pm and 9 pm on Friday night? What is the mean and standard deviation of the number of calls between 10 pm and midnight on Friday night? P(X = 0) = poissonpdf(3.5,0) = P(X = 0) = poissonpdf(7,0) = From 8:00 until 8:30 is a 30 minute period. From 8:00 until 9:00 is a 60 minute period. Since the period is doubled, you must double the mean amount of calls to keep it proportional!  = 14 &  = Be sure to adjust !

Examine the histograms of the Poisson distributions – = 2 = 4 = 6 What happens to the shape? What happens to the means? What happens to the standard deviations?

As increases The distributions become more symmetrical The means increase The standard deviations increase

Assignments ASSIGNMENTS DUE MONDAY –WS Geometric Activity –WS Geometric Distributions –WS Poisson Distributions MONDAY: QUIZ – Geometric & Poisson Distributions NOW: QUIZ – Discrete & Binomial Distributions