1. LESSON 11: THE POISSON DISTRIBUTION
Outline
The context
The properties
Notation
Formula
Use of table
Use of Excel
Mean and variance

2. POISSON PROCESS THE CONTEXT
An important property of the Poisson process:
The probability of two or more successes will occur in an interval approaches zero as the interval becomes smaller.
Examples where interval is a length of time
Number of people coming to a bank per hour
Number of calls received by a telephone operator per hour
Number of accidents per week
Examples where interval is a length of an item
Number of defects per hundred meter

3. POISSON PROCESS THE PROPERTIES
The Poisson process has the following properties:
1. The number of successes of various intervals are independent. A Poisson process has no memory.
2. The probability that a success will occur in an interval is the same for all intervals of equal size and is proportional to the size of the interval. The mean process rate  must remain constant for the entire time span pr space considered.
3. The probability of two or more successes will occur in an interval approaches zero as the interval becomes smaller.

4. POISSON DISTRIBUTION THE NOTATION
 : the mean rate of success per period
t : the time span (or space) considered
x : the number of observed successes
A constant
The Excel function for ex is EXP(x)

5. POISSON DISTRIBUTION THE PROBABILITY DISTRIBUTION
The Poisson probability distribution gives the probability of getting exactly x successes during a given time interval or in a specified region
The probability of getting exactly x successes during a given time interval or in a specified region is as follows:

6. POISSON DISTRIBUTION THE PROBABILITY DISTRIBUTION
Example 1: The marketing manager of a company has noted that she usually receives 15 complaint calls from customers during a week (consisting of 5 working days) and that the calls occur at random. Find the probability of her receiving exactly 5 calls in a single day.

7. POISSON DISTRIBUTION THE PROBABILITY DISTRIBUTION
Example 2: The marketing manager of a company has noted that she usually receives 15 complaint calls from customers during a week (consisting of 5 working days) and that the calls occur at random. Find the probability of her receiving at most 2 calls in a single day.

8. POISSON DISTRIBUTION USE OF TABLE
Table C, Appendix A, pp gives the probability of getting at most x successes given that t is the average number of successes occurring in the given time interval or region
The table can be used to find the probability of
exactly x successes:
at least x successes:
successes between a and b:

9. POISSON DISTRIBUTION USE OF TABLE
Example 3: Find the following using Table C:
Example 4: Find the following using Table C:

10. POISSON DISTRIBUTION USE OF EXCEL
The Excel function POISSON gives and
It takes three arguments, the first 2 being x and t
The last one is TRUE for and FALSE for
Example 5 (self study): Find and Verify if Excel gives the same answer as it was obtained before.
Answer:
= POISSON(2,3,TRUE)
= POISSON(2,3,FALSE)

11. POISSON DISTRIBUTION MEAN AND VARIANCE
If X is a Poisson random variable, the mean and the variance of X are:
E(X) is the mean or expected value of X
V(X) is the variance of X
 is the mean rate of successes per period
t is the length of time (or space) considerd

12. READING AND EXERCISES Lesson 11 Reading: Section 8-1, pp. 228-233
8-1, 8-2