The Poisson PD

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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 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-??X/X!, where ? 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. Table E.7 has information about various
mean amounts during a interval. Since our ? 3
we look in that column X P(X) 0 .0498 1 .1494 2
.2240 And so on. On the next slide I
have produced the P(X) column in Excel and I have
put in the cumulative probability column.
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What does the Cum 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(X5)? So,
P(x5) .9161. What about P(x5)? This equals 1
- P(x5) .0839 Note on the previous screen that
the Cum 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 cum.
prob. You could just use the table in the book.
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Problem 22 page 170
From Table E.7 column 6.0 X P(X) 0 .0025 a. P(X
P(4) .2851 2 .0446 b. P(5) .1606 3 .0892 c.
P(X5) 1 P(X P(4) P(5) .2945 5 .1606 6 .1377 question
37 page 201 And so on. Discard if 3 or less
P(X3) P(0) P(1) P(2) P(3)
.1512 They can expect to discard .1512(100)
15.12 cookies