SOCR EduMaterials Activities PoissonExperiment

Description

The experiment is to run a Poisson process until time t. The arrivals are shown as red dots on a timeline, and the number of arrivals N is recorded on each update. The density and moments of N are shown in the distribution graph and the distribution table. The parameters of the experiment are the rate of the process r and the time t, which can be varied with the scroll bars above.

Goal

To provide a simulation demonstrating the effects when varying the time and the rate of the process of interest, and to give a better generalization of the Poisson distribution analytically and graphically.

Experiment

Go to the SOCR Experiment and select the Poisson Experiment from the drop-down list of experiments on the top left. The image below shows the initial view of this experiment:

When pressing the play button, one trial will be executed and recorded in the distribution table below. The fast forward button symbolizes the nth number of trials to be executed each time. The stop button ceases any activity and is helpful when the experimenter chooses “continuous,” indicating an infinite number of events. The fourth button will reset the entire experiment, deleting all previous information and data collected. The “update” scroll indicates nth number of trials (1, 10, 100, or 1000) performed when selecting the fast forward button and the “stop” scroll indicates the maximum number of trials in the experiment.

Because parameter t and r may be varied, both variables are able to modify the shape of the distribution graph. When either r or t increase, the distribution graph takes a more accurate shape of the normal curve, whereas decreasing the values of these two variables will cause the graph to be less normal. The images shown below demonstrate these effect as the first illustrates what happens when t is increased and the second illustrates what happens when both t and r are increased:

Note that the empirical density and moments graph begin to converge to the distribution graph after every trial. As shown:

Applications

The Poisson Experiment is an applet that generalizes the importance of experiments involving Poisson distributions. It may be used in many different types of examples:

Suppose one car leaves a large parking structure every five minutes. A car leaves the parking structure and a space is available. You want to park but another car takes the space. You want to determine when the next spot will be available if a car leaves the parking structure every five minutes while a car enters the parking structure every ten minutes.

Suppose there is a tank in which water leaks out at a rate of five gallons per minute. At the same time, a solution with 5% pollution enters the tank. Engineers want to be able to illustrate this event by using the Java applet for Poisson distribution.

References

• See the Poisson lecture notes.

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