1. Poisson Probability Distributions
Section 5.5
Poisson Probability Distributions

2. Poisson Probability Distribution
What is it?
Requirements:
A poisson probability distribution is a special kind of 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).
The random variable x is the number of occurrences of an event 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 being used.

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

4. A formula for Calculating Probability
The Formula
The Components
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.
mu: The mean.

5. Differences Between Binomial and Poisson
The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean, mu.
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, …, with no upper limit.

6. 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 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?
Binomial! The values of x are limited, not over a range of time or any period (over a range of n = 6)

7. 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

8. 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
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

9. 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.
Poisson: .253; binom: .254

10. 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.
Find the probability that on a given day, there are more than 2 motor vehicle deaths.
Is it unusual to have more than 2?
.0970
1 – P(0) – P(1) – P(2)
/- 2(.311) = [.622 , .720]; Or, yes because P(more than 2) </= .05

11. Other Options
TI-83+ and TI-84 Direction
Your Toolbox
Press 2nd VARS (to get DISTR), then select the option identified as poissonpdf
Complete the entry of poissonpdf( , x) then press ENTER.
Use a TI-83/84 Plus.
Use the formulas

12. 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.
Find the probability that on a given day, there are more than 2 motor vehicle deaths.
Is it unusual to have more than 2?

13. 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.
Find the probability that on a given day, 50 radioactive atoms decayed.
62.2
.0156

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