Transcript The Poisson Distribution - Mr Santowski`s Math Page

6.10 - Discrete Random
Variables: The Poisson
Distribution
IBHLY1/Y2 - Santowski
(A) The Poisson Distribution
the Poisson distribution resembles the binomial distribution in
that is models
counts of events.
for example, a Poisson distribution could be used to model the
number of accidents at an intersection in a week  however, if
we want to use the binomial distribution, we have to know both
the number of people who make enter the intersection, and the
number of people who have an accident at the intersection,
whereas the number of accidents is sufficient for
applying the Poisson distribution
(A) The Poisson Distribution
the Poisson distribution is an appropriate model for counting
certain types of data  examples of such data are mortality of
infants in a city, the number of misprints in a book, the number
of bacteria on a plate, and the number of activations of a Geiger
counter  key idea here being that the data being counted
happens rarely
although the data happens rarely, we still wish to be able to say
something about the nature of these events
here then, the Poisson distribution can be a useful tool to
answer questions about these rare events
(B) The Poisson Distribution Formula
as with other distribution models, the Poisson distribution is a
mathematical rule that assigns probabilities to the number
occurrences
the only thing we have to know to specify the Poisson
distribution is the mean number of occurrences for which the
symbol lambda () is often used  i.e. the mean, E(X), is  and
interestingly enough, the Var(X) is also equal to , so then the
SD(X) = sqr()
the probability density function of a Poisson variable is given by
e   x
P ( X  x )  px 
, x  0,1,2,3,4,5,....
x!
where again  =  and represents the mean number of
occurrences
(C) Conditions for Poisson
Distributions
Conditions under which a Poisson distribution holds



counts of rare events
all events are independent
average rate does not change over the period of interest
Examples of experiments where a Poisson distribution holds:



birth defects
number of sample defects on a car
number of typographical errors on a page
- one other note ==> the Poisson distribution will resemble a
binomial distribution if the probability of an event is very small
(D) Example
The number of patients admitted in a 24 hour period in an emergency
clinic requiring treatment for gunshot wounds is presented in the table
below.
(i) Find the mean of the distribution
(ii) Compare the actual data with that generated by the Poisson model
X
0
1
2
3
4
5
6
frequency
13
27
27
19
9
4
1
(D) Example
First, we will use the GDC (or we can use our skills and knowledge
from chap 18 – Stats) to find the mean
Using the GDC, we will use L1 and L2 to enter the data and then do a
1-Var Stats calculation to find the mean  which is 2 and that we have
100 data points
Now, to compare our data to that of the theoretical data generated by
a Poisson distribution, we use our GDC again
From the DISTR menu, select the poissonpdf( command
Syntax is poissonpdf(mean, values of X i.e. L1)
Multiply by 100 (# of data points)
Then store the resultant list of values in L3  use STO L3
Now go to L3 to see the listed results
(D) Example
The results are as follows:
X
0
1
2
3
4
5
6
Frequency
13
27
27
19
9
4
1
Poisson
Distribution
13.534
27.067
27.067
18.045
9.0224
3.6089
1.203
So the distribution does show a Poisson distribution
(D) Example
Now we can do the same problem algebraically using
the Poisson distribution formula  once we know the
mean (which can calculate with or without a GDC)
The equation is P(X=x) = (2x)(e-2)/x!
Now we can sub in specific values for x =
{0,1,2,3,4,5,6}
Or we can graph it and generate a table of values for
x = {0,1,2,3,4,5,6}
(D) Example – Graph & Table
The table of values is:
x
0.00000
1.00000
2.00000
3.00000
4.00000
5.00000
6.00000
7.00000
y
0.13534
0.27067
0.27067
0.18045
0.09022
0.03609
0.01203
0.00344
(E) Internet Links
Poisson distribution from C. Schwarz at SFU
Poisson Distribution from the Engineering
Statistics Handbook from NIST
Poisson distribution from Stattucino webbased Statistics Analysis
(F) Classwork
HL math text, Chap 30H, p747, Q1-5
plus other resources