Poisson and uniform paradox

  • #1

Main Question or Discussion Point

Hi, all,

Let's say we deploy some random points on a line of finite length according to a poisson distribution of density \lambda. Can I say that these points are also "uniformly" distributed on the same line?

thks
 

Answers and Replies

  • #2
HallsofIvy
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Hi, all,

Let's say we deploy some random points on a line of finite length according to a poisson distribution of density \lambda. Can I say that these points are also "uniformly" distributed on the same line?

thks
No, you can't. (And there is no "paradox".) With Poisson distribution, with parameter [itex]\lambda[/itex], points are as likely to be [itex]\le \lamba[/itex] as larger. With a "uniform" distribution, they are as likely to be less than or equal to the midpoint of the interval as to be above it. Further, points in a Poisson distribution are more likely to be close to \lambda than not while there is no number in an interval that points in a uniform distribution are more likely to be close to.
 
  • #3
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No, you can't. (And there is no "paradox".) With Poisson distribution, with parameter [itex]\lambda[/itex], points are as likely to be [itex]\le \lamba[/itex] as larger. With a "uniform" distribution, they are as likely to be less than or equal to the midpoint of the interval as to be above it. Further, points in a Poisson distribution are more likely to be close to \lambda than not while there is no number in an interval that points in a uniform distribution are more likely to be close to.
The Poisson distribution gives the waiting time until the next event. I think he means, distribute points randomly along an interval such that their _waiting times_ are distributed as Poisson random variables.

The resulting distribution of points is similar to what you would get if you sampled the same number of points from a uniform distribution along the same interval. There's at least one important difference, however: if you choose points with waiting times according to the Poisson distribution, you don't know when starting out how many points are going to fit in the interval.
 
  • #4
gel
533
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Let's say we deploy some random points on a line of finite length according to a poisson distribution of density \lambda.
That doesn't make sense, the Poisson distribution is unbounded, so not confined to a random line. Maybe you mean a random (and finite) set so that the number of points in any interval (a,b) is Poisson distributed with parameter [itex]\lambda(b-a)[/itex]? i.e., a http://books.google.co.uk/books?id=...bSNCw&sa=X&oi=book_result&ct=result&resnum=5".
Then, yes, you get the same thing as choosing N independent and uniform random variables, where N itself has the Poisson distribution.

(I don't see any paradox...)
 
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  • #5
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To be more precise, for a Poisson process (where the interarrival times are exponential) given that there are N arrivals in [0,T] the arrival times T1,...,TN will be distributed according to the order statistics of N independent uniform variables.
 

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