poisson and uniform paradox

**benjaminmar8** · Jun23-09, 09:37 AM

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

**HallsofIvy** · Jun23-09, 05:51 PM

Originally Posted by benjaminmar8

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 $LaTeX Code: \\lambda$ , points are as likely to be $LaTeX Code: \\le \\lamba$ 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.

**mXSCNT** · Jun23-09, 06:02 PM

Originally Posted by HallsofIvy

No, you can't. (And there is no "paradox".) With Poisson distribution, with parameter $LaTeX Code: \\lambda$ , points are as likely to be $LaTeX Code: \\le \\lamba$ 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.

**gel** · Jun30-09, 06:09 PM

Originally Posted by benjaminmar8

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 $LaTeX Code: \\lambda(b-a)$ ? i.e., a Poisson random field or Poisson random measure.
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...)

**bpet** · Jul2-09, 09:53 PM

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.