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## Thursday, May 22, 2014

### Poisson basics #2

Derived Rigorously from binomial when the number of coins (N) is large. Note sample size N has completely dropped out of the probability function. See Wolfram.

Note the rigorous derivation doesn't require p (i.e. probability of head) to be small. However, Poisson is useful mostly for small p. See book1 - law of improbable events.

Only for small values of p, the Poisson Distribution can simulate the Binomial Distribution, and it is much easier to compute than the Binomial. See umass and book1.

Actually, It is only with rare events (i.e. small p, NOT small r) that Poisson can successfully mimic the Binomial Distribution. For larger values of p, the Normal Distribution gives a better approximation to the Binomial. See umass.

Poisson is applicable where the interval may be time, distance, area or volume, but let's focus on time for now. Therefore we say "Poisson Process". The length of "interval" is never mentioned in Poisson or binomial distributions. The Poisson distribution vs Poisson process are 2 rather different things and confusing. I think it's like Gaussian distro vs Brownian Motion.

I avoid "lambda" as it's given a new meaning in the Poisson _process_ description -- see HK.

Poisson is discrete meaning the outcome can only be non-negative integers. However, unlike binomial, the highest outcome is not "all 22 coins are heads" but infinite. See book1. From the binomial view point, the number of trials (coins) during even a brief interval is infinitely large.

---Now my focus is estimating occurrences given a time interval of varying length. HK covers this.
I like to think of each Poisson process as a noisegen, characterized by a single parameter "r". If 2 Poisson processes have identical r, then the 2 processes are indistinguishable. In my mind, during each interval, the noisegen throws a large (actually inf.) number of identical coins with small p. This particular noisegen machine is programmed without a constant N or p, but the product of N*p i.e. r is held constant.

Next we look at a process where the r is proportional to the interval length. In this modified noisegen, we look at a given interval, of length t. The noisegen runs once for this interval. The hidden param N is proportional to t, so r is also proportional to t.

References -
http://mathworld.wolfram.com/PoissonDistribution.html - wolfram
http://www.umass.edu/wsp/resources/poisson/ - umass
My book [[stat techniques in biz and econ]] - book1
http://www.math.ust.hk/~maykwok/courses/ma246/04_05/04MA246L4B.pdf - HK