Dan Piett STAT West Virginia University Lecture 6

Last Week Expected Value of Probability Distributions Binomial Distributions Probabilities Mean Standard Deviation

Overview Poisson Distribution Poisson Probabilities

Poisson Distribution Suppose an experiment possesses the following properties: 1. The random variable X counts the number of occurrences of some event of interest in a unit of time or space (or both) 2. The events occur randomly 3. The mean number events per unit of space/time is constant 4. The random variable X has no fixed upper limit (This will usually be false, but assume that if there is an upper limit, it is reasonably large) This is a Poisson experiment (NOT pronounced Poison) Note that Poisson Distributions are Discrete (You cannot have occurrences)

Example: Number of Pieces of Mail in a Day (Mean of 5 per day) Requirements This Experiment 1. Random variable X counts the number of occurrences of some event of interest in a unit of time or space (or both) 2. The events occur randomly 3. The mean number events per unit of space/time is constant (lambda) 4. The random variable X has no fixed upper limit 1. Random variable X counts the number of pieces of mail occurring in 1 day. 2. We can assume that the pieces of mail occur randomly 3. Mean number of events per day is constant ( We can assume this as long as we assume that this mean holds true for the times we are interested in) 4. There is no upper-limit to the mail you can receive in 1 day (Debatable, but immaterial)

General Poisson Distribution Suppose X counts the number of occurrences in a Poisson experiment. Then X follows a Poisson Distribution Notation: Pois stands for Poisson distribution lambda stands for the mean number of occurrences per unit time/space For the previous example X~Pois(5)

Formula for a Poisson Distribution Probability of Success in a Poisson Distribution

Problem on Board Assume the mean number of people who visit the emergency room of a particular hospital is 6 per hour. Does this constitute a Poisson Experiment? Find The prob that 0 people will visit the emergency room in 1 hour. The prob that 7 people will visit the emergency room in1 hour.

Cumulative Poisson Probabilities The previous formula can be used to find the probability that X equal to exactly some value What about other probabilities of interest? X equal to less than some value? X equal to more than some value? X is between two values? How do we do this? EXACTLY like Binomial Probabilities

Back to the Previous Example What is the probability that at most 3 people visit the emergency room in 1 hour? At most = less than or equal to At most 3 people= {0, 1, 2, 3, 4,…} P(At most 3 people) = P(X=0)+P(X=1) + P(X=2)+P(X=3) Note: The probability of this event is defined as the sums of the probabilities. Remember that this only works because Poisson Probabilities are discrete Looking pretty familiar? I’m sure you can guess an easier way.

New Example Suppose that an archaeologist finds artifacts at a dig site at an average rate of 2 per day. What is the probability that the archaeologist finds fewer than 4 artifacts in a day. Fewer than 4= {0, 1, 2, 3, 4, 5, …} P(3 or less) = P(X=0) + P(X=1) + P(X=2) + P(X=3) We would need to compute 4 probabilities to solve this. Is there a better way? Unlike Binomial, we only have 1 alternative method Using cumulative probability tables Why doesn’t the complementary rule work for less than probabilities?

Cumulative Probability Tables Because of the difficulty of calculating these probabilities (and how common the poisson distribution is). Cumulative probabilities for specific values of lambda and x have been tabulated. Note: These tables will be provided on exams. How to read the table: Find the appropriate lambda value, look for x This is the probability that X is less than or equal to that value

Back to our Example We have our archaeologist finding 2 artifacts on average per day. What is the probability that he: Finds at most 1 artifact? Finds less than 7 artifacts

Greater than Probabilities So we now know how to calculate the probability that X is equal to exactly some value or the probability that X is less than/less than or equal to some value. What about the probability that X is greater than/greater than or equal to some value? Think back to complementary probabilities

Headed back to our Example Suppose the archaeologist moves to a new dig site and can find an average of 4 artifacts per day now. What is the probability that he finds 5 or more artifacts? 5 or more = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …} P(5 or more) = P(5) + P(6) + … P(infinity) Remember back to how we handled this with binomial probabilities P(5 or more) and P(4 or less) are complementary events What does this mean? P(5 or more) = 1 – P(4 or less)

Greater than Probabilities Remember back to our use of the tables for calculating less than or equal to probabilities We can likewise calculate greater than/greater than or equal to probabilities using the table. Watch the = We want to get our greater than probabilities in terms of less than or equal to P(X>3) = 1 – P(X<=3) {1, 2, 3, 4, 5, …) P(X>=3) = 1 – P(X<3) = P(X<=2) {1,2,3, 4, 5, 6}

In-between Probabilities So far we’ve done P(X=x), P(X x) One more to go (The probability the X is between 2 values) P(a < =x <= b) Example: P(X is between 2 and 6) Between 2 and 6 = {0, 1, 2, 3, 4, 5, 6, 7, … ) P(X is between 2 and 6) = P(X<=6) – P(X<=1) Why? P(X<=6) = P(0) + P(1) + P(2)+P(3)+P(4)+P(5)+P(6) P(X<=1) = P(0) + P(1) Subtract these and the 0 and 1 cancel leaving: P(2)+P(3)+P(4)+P(5)+P(6) This is what we want

Coming back to Exact Probabilities We can use the cumulative table to find exact probability as well P(X=2) = P(X<=2) – P(X<=1) Same logic as the previous examples P(X<=2) = P(X=0) + P(X=1) + P(X=2) P(X<=1) = P(X=0) + P(X=1) Subtract and you are left with P(X=2)

Mean and Standard Deviation of a Poisson Distribution Mean of a Poisson Distribution lambda Standard Deviation of a Poisson Distribution Sqrt(lambda)