Please break this down for a non-statistition.

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Request for Question Clarification by mathtalk-ga on 06 Dec 2002 18:13 PST

Hi, donkan-ga:

Please clarify what you mean by "calculating" approximate Poisson
distribution probabilities.  Are you looking for something you can do
in your head?  on the back of an envelope?  with a TI calculator in
hand?  Or were you asking about numerical methods for approximating
the Poisson distributions to any desired degree of accuracy?

For a well-targeted answer, perhaps you could also amplify on the
non-statistician situation.  What are your skills and interests in
relation to this exercise?  For example, you might be building an
Excel spreadsheet to predict light bulb lifetimes; you know Excel but
not the statistical application.

regards, mathtalk-ga

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Clarification of Question by donkan-ga on 07 Dec 2002 03:06 PST

mathtalk-ga, sorry for the lack of precision in my question. I just
remember years ago putting such a function in a calculator something
like a TI that could answer questions such as this, which I just made
up:

"Assume that fatal accidents at a busy intersection A are randomly
distributed in time, and an average of 3/year occur. What is the
probability that exactly 5 will occur at A in 2003?"

After seeing your request for clarification I found The Poisson Pit
(http://www.anesi.com/poisson.htm), but I'm not sure I understand
their example. I think their p(x;m) is what I'm after (and it's degree
of accuracy). If I use their p(x;m) function for my example, I think
I'd use m=3, but what is my x?

I first came across the Poisson Distribution many years ago in a
freshman physics class. We were given Geiger counters and asked to
describe the distribution of "cosmic rays/particles" hitting the
counter. The answer I believe was the PD. So let's use this as another
example. Let's say that an average of 3 particles hit the counter
every 10 minutes. How do I get the probability of exactly 30 hits in
the next hour? What are the x and m?

I hope this narrows down my question. Forget the non-statistician
part. I'm not one, obviously, but I think I'll be able to follow your
answer as it pertains to p(x;m).

Hi Donkan-ga,

First, to answer your specific questions: in your first example with
car accidents, you would indeed take m=3, which is the expected number
of fatal accidents in a year. As for x, you would set x=5 to get the
probability p(x;m) of exactly 5 fatal accidents occurring in a given
year in that intersection.

In the second example, m will be equal to 18, since if the expected
number of particle hits every 10 minutes is 3, then in one hour it
will be 3*6=18. x would be 30, since you are interested in the
probability of getting exactly 30 hits.

Now, for a more general explanation: The Poisson distribution
describes the number of "rare events" happening in a given time/space
interval. As you seem to already understand pretty well, the type of
"rare events" can be anything from car accidents, radioactive particle
disintegration, to rain drops hitting a sidewalk or spelling mistakes
in a newspaper page.

The function p(x;m) gives the probability that exactly x events will
happen, provided that m is the expected number of events happening. It
is given by the expression

 p(x;m) = e^(-m) m^x / x! ("^" means "to the power of", "!" means
factorial)

 (x = 0,1,2,3, ...)

Sometimes you know the "m" for a different time or space interval than
that for which you want to compute probabilities. For instance, if I
know that an average page of a certain newspaper contains 5 spelling
mistakes, then the probability that on a given day there will be 7
mistakes on the first two pages will be p(7;10) - I take m=5*2 to
account for the number of pages I look at.

Further information:
Poisson Distribution - from Eric Weisstein's world of Mathematics
http://mathworld.wolfram.com/PoissonDistribution.html

Hope this helps,
dannidin

Outstanding answer. Thank you!