Poisson (+ exponential) distribution

See also post with 4 references including my book, HK, UMass.

Among discrete distributions, Poisson is one of the most practical yet simple models. I now feel Poisson model is closely linked to binomial
* the derivation is based on the simpler binomial model - tossing unfair coin N times
* Poisson can be an approximation to the binomial distribution when the number of coins is large but not infinite. Under infinity, I feel Poisson is the best model.

I believe this is probability, not statistics. However, Poisson is widely used in statistics.

Eg: Suppose I get 2.4 calls per day on average. What's the probability of getting 3 calls tomorrow? Let's evenly divide the period into many (N) small intervals. Start with N = 240 intervals. Within each small interval,

Pr(a=1 call) ~= 1% ( ? i.e. 2.4/240?)
Pr(a=0) = 99%
Pr(a>1) ~= 0%. This approximation is more realistic as N approaches infinity.

The 240 intervals are like 240 independent (unfair) coin flips. Therefore,
Let X=total number of calls in the period. Then as an example

Pr(X = 3 calls) = 240-choose-3 * 1%³ * 99%²³⁷. As N increases from 240 to infinite number of tiny intervals,
Pr(X = 3) = exp(-2.4)2.4³/ 3! or more generically
Pr(X = x) = exp(-2.4)2.4^x/ x!

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Incidentally, there's an exponential distribution underneath/within/at the heart of the Poisson Process (I didn't say Poisson Distro). The "how-long-till-next-occurrence" random variable (denoted T) has an exponential distribution whereby Pr (T > 0.5 days) = exp(-2.4*.5). In contrast to the discrete nature of the Poisson variable, T is a continuous RV with a PDF curve (rather than a histogram). This T variable is rather important in financial math, well covered in the U@C Sep review.

For a credit default model with a constant hazard rate, I think this expo distribution applies. See other posts.