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 occurrencesthe 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

where again = and represents the mean number of occurrences

P X x pe

xxx

x

( )!, , , , , , ,....

0 1 2 3 4 5

(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 meanUsing 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 againFrom the DISTR menu, select the poissonpdf( commandSyntax 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:

So the distribution does show a Poisson distribution

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

(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 y 0.00000 0.13534 1.00000 0.27067 2.00000 0.27067 3.00000 0.18045 4.00000 0.09022 5.00000 0.03609 6.00000 0.01203 7.00000 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 web-based Statistics Analysis

(F) Classwork

HL math text, Chap 30H, p747, Q1-5plus other resources