Poisson and Geometric

Rare events over time, and the wait until the first success.

Two distributions to know cold for any quant interview.

The Poisson $(\lambda)$ distribution models the number of rare independent events in a fixed window. Its PMF is

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots

A neat property: mean and variance are both $\lambda$ . Trade arrivals per minute, typos per page, defaults per year are all classically Poisson.

The Geometric $(p)$ distribution counts the number of independent Bernoulli $(p)$ trials until the first success:

P(X = k) = (1-p)^{k-1} p, \quad k = 1, 2, \dots

with $E[X] = 1/p$ and $\mathrm{Var}(X) = (1-p)/p^2$ .

Geometric is memoryless: given that you've already waited $m$ unsuccessful trials, the additional wait until success is distributed exactly like a fresh Geometric $(p)$ . Past failures carry no information about the future. This property is shared with the continuous-time Exponential distribution.