## POISSON Probability Distribution

The POISSON Probability Function is a discrete probability distribution function used to calculate the number of successes "x" from some given interval where the probability of success on each standard interval is the same. We say "x", the random variable, is distributed POISSON ( ,x).

[ To calculate the probability of random variables which are distributed POISSON we will follow a very similar procedure to the one just described.]

Sample Problem

Let's say we want to find the probability of 4 flaws in a 10 yard piece of material where it is known that this material has an average of only 1 flaw every 5 yards. Here our given interval is 10  yards. Our standard interval is  5  yards, and "x" is 4 .

We first need to calculate our expected value, , for 10 yards. We will use the information given:

(1flaw / 5yards) * 10 yards = 2 flaws. This is our  .We would use the formula: P(x) = [( ^x)(e^(-( )] / x ! with =2 and x=4.

On the calculator we go to the distribution menu by pressing Next Arrow up to "B : poissonpdf ( ".
(left screen will be displayed)

Press (you will see the right screen)

Now we must enter , x in that order so for our problem we enter . Press
.0902235222
will be displayed. This is the probability of 4 flaws in 10 yards of material where it is known that the material has on the average only 1 flaw every 5 yards.

"Poissoncdf (" will give the cumulative probability from 0 through X. (e.g. poissoncdf (2,4) will give the probability of x = 0+1+2+3+4 , where =2)

You should get 0.9473469827 for the cumulative answer.

To see a Distribution Plot of a particular POISSON distribution we will need to enter some data into the LIST.
Recall the discussion from the Binomial Distribution Plot. (see mini-lesson 4)

We will enter the "x" values {1,2,3,4,5,6,7,8,9,10} into L1 and the POISSON Formula into L2 .

Press "EDIT" should be highlighted. Press

Enter the numbers {1,2,3,4,5,6,7,8,9,10} into List1. These are the "x" values. Next we want to enter the formula into List2.

First highlight L2 (you should see lower left screen)

Next enter
LN .
(You should see this text scroll across the screen)

We now need to finish the formula by dividing by L1 factorial.

So press (see lower left screen)

Arrow left to "PRB" and press (see upper screen)

Now press List2 will be filled with these probabilities
(see screen below)

The numbers in List2 are the probabilities associated with the "x" values in List1.

To see a plot go to the STAT PLOT so press
(see screen below)

"1:Plot1" should be highlighted.

Press (see screen)

Make sure On is highlighted and press

Next Arrow down and highlight the plot option. Press

Make sure Xlist: L1 , Ylist: L2

Now, as always, we need to adjust the window to fit our data, or you may press

To adjust the window press Match these settings:

Xmin: 0 , Xmax: 11 ,Xscl 1 , Ymin: 0 ,Ymax: 0.8 , Yscl 0.1 .

Press then ( you will see this screen)

(Notice this graph more closely resembles a negative exponential curve rather than a “bell curve”. )

Here "x" is the number of successes and "y" is the corresponding probability. So from this screen we see that there is a .27067057 probability of 2 flaws from 10 yards of material where there is an average of only 1 flaw every 5 yards. You can Arrow left or right to read each desired probability.

Notice also that in the upper left screen P1: L1,L2 is displayed. This tells us that we are using PLOT1 and the data from LIST1, and LIST2.