1. Poisson Probability Distributions
Section 5.5
Poisson Probability Distributions

2. Learning Targets
The Poisson distribution is another discrete probability distribution which is important because it is often used for describing the behavior of rare events (with small probabilities).

3. Some events are rather rare - they don't happen that often
Some events are rather rare - they don't happen that often. For instance, car accidents are the exception rather than the rule. Still, over a period of time, we can say something about the nature of rare events.
An example is the improvement of traffic safety, where the government wants to know whether seat belts reduce the number of death in car accidents. Here, the Poisson distribution can be a useful tool to answer questions about benefits of seat belt use.

4. Other phenomena that often follow a Poisson distribution are death of infants, the number of misprints in a book, the number of customers arriving, and the number of activations of a Geiger counter.
The distribution was derived by the French mathematician Siméon Poisson in 1837, and the first application was the description of the number of deaths by horse kicking in the Prussian army.

5. Examples of Poisson Distributions
Patients arriving at an emergency room.
Radioactive decay.
Crashes on a highway.
Internet users logging onto a website.

6. Poisson Probability Distribution
What is it?
Requirements:
A poisson probability distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The Poisson distribution is often used for describing rare events (small probabilities).
Parameters
The random variable x is the number of occurrences of an event over some interval.
The interval can be time, distance, area, volume, or same similar unit.
The occurrences must be random.
The occurrences must be independent of each other.
The occurrences must be uniformly distributed over the interval being used.
The mean is µ.
The standard deviation is  = 𝝁 .

7. A formula for Calculating Probability
The Formula
The Components
𝑷 𝒙 = 𝒆 −𝝁 𝝁 𝒙 𝒙!
P(x): The probability of an event occurring x times over an interval.
e: The number “e” where e =
x: The number of occurrences of the event in an interval.
: The mean which indicates the average number of events in the given time interval.

8. Poisson or not? Which of the following are likely to be well modelled by a Poisson distribution? (can check more than one)
Number of duds found when I test four components
The number of heart attacks in Oswego each year
The number of planes landing at O’Hare Airport between 8 and 9am
The number of cars getting punctures on the Route 59 each year
Number of people in the UK flooded out of their home in July

9. Are they Poisson? Answers:
Number of duds found when I test four components
- NO: this is Binomial (it is not the number of independent random events in a continuous interval)
The number of heart attacks in Oswego each year
- YES: large population, no obvious correlations between heart attacks in different people
The number of planes landing at O’Hare Airport between 8 and 9am
NO: 8-9am is rush hour, planes land regularly to land as many as possible (1-2 a minute) – they do not land at random times or they would hit each other!
The number of cars getting punctures on the Route 59 each year
YES (roughly): If punctures are due to tires randomly wearing thin, then expect punctures to happen independently at random But: may not all be independent, e.g. if there is broken glass in one lane
Number of people in the UK flooded out of their home in July
NO: floodings of different homes not at all independent; usually a small number of floods each flood many homes at once, 𝑃 flooded next door flooded ≫𝑃(flooded)

10. Example: Mercy Hospital
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?
 = 6/hour = 3/half-hour, X = 4

11. Example: Telecommunications
Messages arrive at a switching centre at random and at an average rate of 1.2 per second.
Find the probability of 5 messages arriving in a 2-sec interval.
Let X = number of messages arriving in a 2-sec interval.
Then X follows a Poisson distribution, with mean number =1.2×2=2.4
𝑃 5 = 𝑒 −  𝑘 𝑘! = 𝑒 − ! =0.0602

12. Example
Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying cesium-137, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms.
Find the mean number of radioactive atoms that decayed in a day.
 = ( – )/365 =22713/365 =
Find the probability that on a given day, 50 radioactive atoms decayed.
P(50) = poissonpdf( , 50) =

13. In two hours mean number is =𝟐×𝟑=𝟔.
Example Suppose that trucks arrive at a receiving dock with an average arrival rate of 3 per hour. What is the probability exactly 5 trucks will arrive in a two-hour period?
=3, 𝑃 𝑋=5 = 𝑒 − !
=3, 𝑃 𝑋=2.5 = 𝑒 − !
=5, 𝑃 𝑋=6 = 𝑒 − !
=6, 𝑃 𝑋=5 = 𝑒 − !
Reminder: 𝑃 𝑋=𝑘 = 𝑒 −  𝑘 𝑘!
In two hours mean number is =𝟐×𝟑=𝟔.
𝑷 𝑿=𝒌=𝟓 = 𝒆 −  𝒌 𝒌! = 𝒆 −𝟔 𝟔 𝟓 𝟓!

14. Example
Arrivals at a bus-stop follow a Poisson distribution with an average of 4.5 every quarter of an hour.
Calculate the probability of fewer than 3 arrivals in a quarter of an hour.
The probabilities of 0 up to 2 arrivals can be calculated directly from the formula
with  = 4.5
So P(0) = , P(1)= , and P(2)=
So the probability of fewer than 3 arrivals is
P(0) + P(1) + P(2) = =

15. Calculator Shortcut
TI-83+ and TI-84 Direction
Your Toolbox
Press 2nd VARS (to get DISTR), then select the option identified as poissonpdf / poissoncdf
Complete the entry of poissonpdf( µ( ), x) or
poissoncdf( µ( ), x)
then press ENTER.
Use a TI-83/84 Plus.
Use the formulas
BOOM DONE!

16. Calculator Shortcut
“2nd” and “VARS”
Select “poissonpdf” to calculate individual probability value
P(0) = poissonpdf(4.5, 0), P(1) = poissonpdf(4.5, 1),
P(2) = poissonpdf(4.5, 2).
Select “poissoncdf” to calculate the cumulative probability values
P(0) + P(1) + P(2) = poissoncdf(4.5, 2).

17. Example
Dutchess County, New York, has been experiencing a mean of 35.4 motor vehicle deaths each year.
Find the mean number of deaths per day.
 = 35.4/365 =
Find the probability that on a given day, there are more than 2 motor vehicle deaths.
Is it unusual to have more than 2?
It is very unusual to have more than 2 vehicle deaths on a given day.
𝑷 𝒙>𝟐 =𝟏−𝑷 𝒙≤𝟐 =𝟏−𝑷 𝟎 −𝑷 𝟏 −𝑷 𝟐
=𝟏−𝐩𝐨𝐢𝐬𝐬𝐨𝐧𝐜𝐝𝐟 𝟐 =𝟎.𝟎𝟎𝟎𝟏𝟒𝟏𝟒 < 0.05
It is a rare event.

18. More Practice
Consider a Poisson probability distribution with an average number of occurrences of 2 per period.
Write the appropriate Poisson distribution
What is the average number of occurrences in three time periods?
Write the appropriate Poisson function to determine the probability of k occurrences in three time periods.
Compute the probability of two occurrences in one time period.
Compute the probability of six occurrences in three time periods.
Compute the probability of five occurrences in two time periods.

19. Answer
(a)
(b)
(c)
(d)
(e)
(f)

20. Binomial Distribution
Difference from a
Binomial Distribution
The Poisson distribution differs from the binomial distribution in these fundamental ways:
The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean μ.
In a binomial distribution the possible values of the random variable x are 0, 1, n, but a Poisson distribution has possible x values of 0, 1, 2, , with no upper limit.
Page 231 of Elementary Statistics, 10th Edition.

21. Condition of Poisson Distribution as an Approximation of Binomial Distribution:
In Binomial Distribution, if n is large and p is small, then the Binomial distribution with parameters n and p, is well approximated by the Poisson distribution with parameter  = np, i.e. by the Poisson distribution with the same mean.

22. Poisson Rule of Thumb
The Poisson distribution can also be used to approximate the binomial distribution when using the binomial distribution would result in a tedious calculation. However, certain requirements must be met in order to do this. Not all problems can use a Poisson distribution.
Requirements for Using the Poisson Distribution as an Approximation to the Binomial if both
n ≥ 100
np ≤ 10
In this situation, we need a value for mu. That value can be calculated by using the formula present in section 5.4

23. Example
In the Illinois Pick 3 game, you pay $0.50 to select a sequence of three digits, such as 729. If you play this game once every day, find the probability of winning exactly once in 365 days.
n = 365 > 100, p = 1/1000 = 0.001,
and  = np = < 10
P(1) = poissonpdf(0.365, 1) = (Poisson)
P(1) = binompdf(365, 0.001, 1) =
(Binomial)

24. Example
The probability of a certain part failing within ten years is Five million of the parts have been sold so far.
What is the probability that three or more will fail within ten years?
Let X = number failing in ten years, out of 5,000,000;
n = 𝟓𝟎𝟎𝟎𝟎𝟎𝟎>𝟏𝟎𝟎, 𝒑=𝟏𝟎 −𝟔 very small and np = 𝟓𝟎𝟎𝟎𝟎𝟎𝟎 × 𝟏𝟎 −𝟔 =𝟓<𝟏𝟎
Evaluating the Binomial probabilities is rather awkward; better to use the Poisson approximation.
X has approximately Poisson distribution with  =𝒏𝒑=𝟓.
P(Three or more fail) =𝑷 𝑿≥𝟑
=𝟏−𝑷 𝑿=𝟎 −𝑷 𝑿=𝟏 −𝑷(𝑿=𝟐)
=𝟏− 𝒆 −𝟓 𝟓 𝟎 𝟎! − 𝒆 −𝟓 𝟓 𝟏 𝟏! − 𝒆 −𝟓 𝟓 𝟐 𝟐!
=𝟏− poissoncdf(5, 2) =
For such small 𝒑 and large 𝒏 the Poisson approximation is very accurate (exact result is also to three significant figures).

25. A Lucky Die
An experiment involves rolling a die 6 times and counting the number of 2’s that occur. If we calculate the probability of x = 0 occurrences of number 2 using the Poisson distribution we get 0.368, but we get if we use the binomial distribution. Which is the correct probability of getting no 2’s when a die is rolled 6 times? Why is the other probability wrong?

26. Mean and Standard Deviation of the Poisson Probability Distribution
Formulas
Remember:
The mean is
The standard deviation is
Mean is the same as expected value
This is what you would expect the average to be after an infinite number of trials
Standard deviation represents the spread of the data that you would expect to see

27. Recap In this section we have discussed:
Definition of the Poisson distribution.
Requirements for the Poisson distribution.
Difference between a Poisson distribution and a binomial distribution.
Poisson approximation to the binomial.

28. Homework
Pg #10, 12, 15 (use Binomial and Poisson distributions for #15)