## [hashcash] Re: geometric vs poisson (Re: Statistics of hashcash collisions)

• From: Hubert Chan <hubert@xxxxxxxxx>
• To: hashcash@xxxxxxxxxxxxx
• Date: Thu, 27 May 2004 13:25:34 -0400

```>>>>> "Adam" == Adam Back <adam@xxxxxxxxxxxxxxx> writes:

[...]

[...]

Adam> I think poisson will approximate to q^n rather than q^n-1,
^^^^^^^
Adam> obviously as b and so n=2^b gets larger (recall p=1/n,q=1-p) they

I think you misunderstood me.  To approximate a geometric distribution,
you use an exponential distribution, not Poisson.  Poisson approximates
a binomial distribution.  (And with the probabilities that we're
working with, the approximation should be fairly good.)

Poisson gives you the probability of a given number of hits after a
fixed time (P_v(n) -- v is supposed to be fixed, not variable, in a
Poisson distribution).  Exponential gives the probability that the first
hit occurs within a given time (P(X<=x) where X is the random variable,
and x is the time in question).

The analysis seems to be correct.  It's just the naming that Anonymous
uses is off.

http://mathworld.wolfram.com/GeometricDistribution.html
http://mathworld.wolfram.com/ExponentialDistribution.html
http://mathworld.wolfram.com/BinomialDistribution.html
http://mathworld.wolfram.com/PoissonDistribution.html

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Hubert Chan <hubert@xxxxxxxxx> - http://www.uhoreg.ca/
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```