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Poisson and Geometric Distributions

Poisson: how many events in a window

The Poisson distribution counts events that occur independently at a constant average rate λ\lambda over a fixed interval:

The Poisson distribution
P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Counts of rare events in a fixed window, with the mean and the variance both equal to lambda.

Its defining feature is that mean and variance are the same number:

E[X]=Var(X)=λE[X] = \text{Var}(X) = \lambda

That equality is a testable prediction rather than a convenience. If you count trades per minute and find the variance well above the mean, the arrivals are not Poisson: they are overdispersed, which is the statistical signature of clustering. Order flow is famously overdispersed, because trades arrive in bursts around news rather than at a steady trickle.

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