0012: "Poisson"

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cyanyoshi
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Joined: Thu Sep 23, 2010 3:30 am UTC

0012: "Poisson"

Title text:
Spoiler:
Poisson distributions have no value over negative numbers
Oh yeah, well i am an imaginary number.

rmsgrey
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Re: 0012: "Poisson"

There's something fishy about his claim...

orthogon
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Re: 0012: "Poisson"

These days Randall wouldn't dream of insulting his readers' intelligence by using the title text to explain one of the references. (Not directly, at least, though sometimes the title text gives an additional clue that helps you get the joke).

Does the Poisson distribution really have no value over the negative numbers, i.e. it's undefined, or is it just zero?
xtifr wrote:... and orthogon merely sounds undecided.

Flumble
Yes Man
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Re: 0012: "Poisson"

orthogon wrote:Does the Poisson distribution really have no value over the negative numbers, i.e. it's undefined, or is it just zero?

Assuming the definition
Since the Poisson distribution is discrete (for k) and has a factorial, it is indeed undefined for negative values.

And if Randall was refering to the parameter (λ): Poisson distributions are unheard of for non-positive λs. Maybe they have their use in quantum mechanics. It wouldn't be the first time QM used some rediculous analytic continuation.

cyanyoshi
Posts: 412
Joined: Thu Sep 23, 2010 3:30 am UTC

Re: 0012: "Poisson"

orthogon wrote:Does the Poisson distribution really have no value over the negative numbers, i.e. it's undefined, or is it just zero?

This reminds me of the old question "If a tree falls in a forest and no one is around to hear it, does it make a sound?" Except in this case, it's "If you only ever care about what a distribution P(x;mu) does for nonnegative integers x and mu, what value should it take elsewhere?" There's not really any harm in defining the function to be zero everywhere else [e.g. you could say that P(-2.4;banana) = 0], but that is usually not helpful. It depends on the context.

I get mail in an unusually random manner with 5 letters per day on average. How likely am I to receive -2 letters tomorrow? Depending on who I ask, they could say that there is no such thing as receiving a negative number of letters, and so my question doesn't make sense. Someone else might say that the probability is zero because it is certain that I will not receive a negative number of letters. In the world of mathematics, the former point of view is much more common.

ucim
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Re: 0012: "Poisson"

cyanyoshi wrote:How likely am I to receive -2 letters tomorrow?
If you got seven letters but two were not addressed to you, and the next day you got no letters, but put the misdelivered letters into the right neighbors' boxes, how would you account for it?

There are many ways you could do it, but one sensible way is to extend the idea of "number of letters received" to include negative numbers.

Jose
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cyanyoshi
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Re: 0012: "Poisson"

ucim wrote:
cyanyoshi wrote:How likely am I to receive -2 letters tomorrow?
If you got seven letters but two were not addressed to you, and the next day you got no letters, but put the misdelivered letters into the right neighbors' boxes, how would you account for it?

There are many ways you could do it, but one sensible way is to extend the idea of "number of letters received" to include negative numbers.

Jose

Yes, that is a perfectly reasonable way to count letters. I suppose one could model the number of correctly- and incorrectly-delivered letters as separate Poisson-like distributions in the nonnegative and nonpositive integers, respectively, and perform a discrete convolution. The resulting distribution would certainly not be Poisson, but in this case it would actually be helpful to consider the value these extended Poisson distributions take outside their usual intervals.