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INTRODUCTION TO STATISTICS SECTION 5 THE POISSON DISTRIBUTION |
The
Poisson Distribution
So far we have seen the following discrete distributions
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Distribution |
When to use |
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Binomial |
1. Fixed number of trials 2. Independent 3. Two outcomes 4. Probability remains constant |
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Hypergeometric |
1. Fixed number of trials 2. Two outcomes |
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Geometric |
1. Independent 2. Two outcomes 3. Probability remains constant |
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Multinomial |
1. More than 2 outcomes 2. Independent 3. Fixed number of trials 4. Probability remains constant |
There is another discrete distribution that is useful that applies to events over a specific interval. It is called the Poisson Distribution
Definition
The Poisson Distribution is a discrete probability distribution that applies to occurrences of some event over a specific 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. The probability of the event occurring x times over the interval is given by
Where e is a constant approximately equal to 2.71828..
To use the Poisson distribution you need to have:
- The random variable X as the number of occurrences over some interval
- The occurrences must be random
- The occurrences must be independent of each other
- The occurrences must be uniformly distributed over the interval
For example: The number of planes arriving on time at a specific airport on a given day
Or the number of computers returned to the manufacturer for defective parts during the first year or the number of homicides in a given city over a given year
The Poisson distribution has the following parameters
The mean is 
The standard deviation is 
Note that unlike the Binomial - Poisson distribution does not need a fixed number of trials - in fact it has nothing to do with n at all!
Here are some examples:
Example 1: Mr. Wildman has done a study to determine the number of students that attend office hours. After studying the problem over one month, he determines that on average two students arrive for every office hour he schedules. Find the probability that for a randomly selected office hour, the number of student arrivals is:
a. 0
b. 2
c. 5
d. 9
Solution: Poisson applies since the distribution is over a given time period (an office hour). The mean as stated in the problem is 2. So using the formula:
So: 
Do these solutions make sense to you in terms of the problem?
Of course the TI-83 has a quick and easy method of doing this on the DISTR menu.
Here is a solution to part c of the problem above using the TI-83
Use 2nd VARS to get the DISTR menu
Scroll down to get item B - poissonpdf
Hit enter - Now you enter the information in the following order: the mean first and then the value of x. Here is the screen:
Now hit enter to get the answer - WALA!
Some more examples:
A trucking company operates a large fleet of trucks and last year they had 103 breakdowns. Find the mean number of breakdowns per day and the probability on any given day you have 2 breakdowns or say at least 2 breakdowns
Solution: You had 103 breakdowns all year and so the mean number of breakdowns per day is 103/365 = .282. Using Poisson - you get
P(2) = 
For at least one - you can use poissoncdf on the TI-83
poissoncdf(.282,1) will give you the probability of less than or equal to 1 breakdown, so
1- poissoncdf(.282,1) will give you the probability of 2 or more or 0.0330619
The poisson distribution is sometimes used to approximate the binomial when n is large and p is small. Consider the following which is problem 14 on page 220
If you bet a 7 on a roulette wheel there is a probability of 1/38 of winning. Assume bets are placed on the number 7 in each of 500 different spins
Solution: The mean number of wins is n x p = 500 x 1/38 = 13.2
We use Poisson to get the probability that 7 occurs exactly 13 times
Using the binomial formula: 
As you can see the probabilities are very close