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The Poisson PD

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Are you familiar with the rock group Aerosmith? They have a song called Toys in the Attic. If they had a song called Toys on the Roof, then the “Toys on” part rhymes with the name Poisson when you say “Toys on” real fast. Can you say Poisson?

OKAY, I know this is a silly way to start this section, but I wanted to have you be able to say Poisson correctly and “Toys on” said real fast is all I could come up with.

The Poisson PD is useful when certain types of problems are encountered. Some examples might be the number of people who arrive at a Dairy Queen each hour, or the number of phone calls that come into a bank card center each fifteen minute interval. The number of occurrences is a random variable that can be described by the discrete Poisson PD.

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If, in an experiment,

1. the probability of an occurrence is the same for any two intervals of equal length, and

2. the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval,

then the number of occurrences can be described by the Poisson PD. If x is the number of occurrences in the interval, then P(X) is the probability of x occurrences.

Now P(X) = e-μµx/x!,

where μ = the expected value or mean number of occurrences, and

e = 2.71828

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Example: Say we are interested in the number of arrivals at a drive-up teller window during a 1-minute period during the noon hour. Say from past experience we know that on average 3 cars drive up to the window in a 1 minute period.

On the next slide I have produced a table in Excel and I have put in the cumulative probability column last. The table in the back of our book starting on page 307 has the cumulative probability. Please note the similarity between the book and what I have on the next screen.

Now, on the next slide if I have 2 as the number of arrivals (when on average we have 3), then the number in the cumulative column refers to the probability of 0 or 1 or 2 arrivals. If I just want the probability that 2 arrive from the cumulative column take the cumulative value at 2 and subtract the cumulative value at 1.

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Mean Number of Occurrences 3

Number of Arrivals P(X)

Cumulative Probability

0 0.049787068 0.0497870681 0.149361205 0.1991482732 0.224041808 0.4231900813 0.224041808 0.6472318894 0.168031356 0.8152632455 0.100818813 0.9160820586 0.050409407 0.9664914657 0.021604031 0.9880954968 0.008101512 0.9961970089 0.002700504 0.998897512

10 0.000810151 0.99970766311 0.00022095 0.99992861312 5.52376E-05 0.999983851

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What does the Cumulative Prob on the previous screen do for us? It helps us answer questions like what is the probability that 5 or fewer arrivals occur in the interval? We might write this

P(X≤5)?

So, P(x≤5) = .9161. What about P(x>5)? This equals 1 - P(x≤5) = .0839

Note on the previous screen that the Cumulative Prob does not equal 1 when x = 12. The reason is that I just listed up to 12 arrivals, but I could have listed more.

In problems in this section they do something peculiar. They might say something like on average 48 calls arrive per hour. How many calls on average come in 5 minutes? Since 5 minutes is 1/12 of an hour, divide the 48 by 12, or 4. IN OTHER WORDS, pay attention to the time interval and average occurrences per time interval.

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In excel the Poisson function has the form

=poisson(number of arrivals, mean number of arrivals, false or true)

The number of arrivals is in the x column, the mean number of arrivals is at the top of the table, and use false if you want P(X) or use true if you want the cumulative prob.

You could just use the table in the book.