Poisson Distribution

Motivation: why this matters in quant finance

The Poisson distribution counts rare events over a fixed exposure: defaults in a year, trades in a minute, claims in a month, or jumps in an asset price over a day. It is the count side of the same modelling world in which the exponential distribution describes waiting times.

Its key origin is the rare-event limit. Start with many independent Bernoulli opportunities, each with tiny probability, and keep the expected count stable. The binomial count becomes approximately Poisson. Mosteller's counterfeiter and mold-colony problems teach exactly this modelling move.

Definition

A nonnegative integer-valued random variable

N

follows

\operatorname{Pois}(\lambda)

\mathbb{P}(N=k)=e^{-\lambda}\frac{\lambda^k}{k!}, \qquad k=0,1,2,\ldots.

The parameter $\lambda$ is the expected count in the window.

Key Properties

For $N\sim\operatorname{Pois}(\lambda)$ ,

\mathbb{E}[N]=\lambda, \qquad \operatorname{Var}(N)=\lambda.

This equality is useful and restrictive. If observed counts have variance far larger than the mean, the pure Poisson model is probably missing clustering or random intensity.

If $X_n\sim\operatorname{Bin}(n,p_n)$ and $np_n\to\lambda$ , then

X_n\xrightarrow{d}\operatorname{Pois}(\lambda).

If independent $N_i\sim\operatorname{Pois}(\lambda_i)$ , then

\sum_iN_i\sim\operatorname{Pois}\left(\sum_i\lambda_i\right).

The MGF and probability generating function are

M_N(t)=\exp(\lambda(e^t-1)), \qquad G_N(z)=\exp(\lambda(z-1)).

In Quant Finance

In Merton jump-diffusion, the number of jumps by time

T

N_T\sim\operatorname{Pois}(\lambda T).

In credit portfolios, independent low-probability defaults can be approximated by Poisson counts. In market microstructure, trade or quote arrivals are sometimes modelled as Poisson at a first pass.

ISL's Poisson regression framing is important for applied quants: when count intensity depends on predictors, the distribution supplies the likelihood and the regression model supplies the intensity.

Worked Example: Default Count Tail

If $D\sim\operatorname{Pois}(5)$ , then

\mathbb{P}(D\geq10)=1-\sum_{k=0}^{9}e^{-5}\frac{5^k}{k!}\approx0.0318.

That is a capital-relevant tail probability, but it assumes independent, constant-intensity defaults. Common macro factors can make the true tail much larger.

Common Confusions and Pitfalls

$\lambda$ is a probability. It is an expected count;

\mathbb{P}(N=0)=e^{-\lambda}

Poisson automatically handles crisis clustering. It does not. Mixtures, Cox processes, factor models, or self-exciting processes are needed for clustered arrivals.

Mean equals variance in all count data. Real count data are often over-dispersed.

Where This Goes Next

Exponential Distribution: waiting times between Poisson arrivals are exponential.
Bernoulli and Binomial Distributions: the Poisson is the rare-event limit of binomial counts.
Moment-Generating Functions: transforms prove additivity and compound-loss formulas.

References

Bertsekas and Tsitsiklis, Introduction to Probability, 2nd ed., Ch. 2 Sec. 2.2, Sec. 2.4, Sec. 2.7 and Ch. 4 Sec. 4.4.
Mosteller, Fifty Challenging Problems in Probability, Problems 27-30, for Poisson-limit reasoning and count examples.
James, Witten, Hastie, and Tibshirani, An Introduction to Statistical Learning, 2nd ed., Ch. 4 Sec. 4.6-4.7, for Poisson regression.