Poisson distribution

The Poisson distribution models the number of events over an interval (time, area, volume) when events are rare, independent, and occur at a constant rate . It arises as the limit of the binomial and as the jump law of a Poisson process.

Characteristics

Characteristic	Formula	Description
Notation		Sometimes or
Parameter		Mean rate (intensity): expected number of events over the interval
Support		Non-negative integers, countably infinite support
Probability mass function		For
Cumulative distribution function		Also (upper incomplete gamma)
Moment-generating function		Derivatives at 0 yield moments
Characteristic function		Fourier transform of the law
Mean		Equals the intensity parameter
Variance		Equidispersion: mean = variance (diagnostic property)
Standard deviation		Relative fluctuations of order
Mode		Two modes and if is a positive integer
Median	or	Common approximation: for large
Skewness		Right-skewed; tends to 0 as
Excess kurtosis		Heavier tails than the normal for small
Shannon entropy	—	No simple closed form; numerical computation via the PMF
Fisher information		For one observation; decreases with intensity
Kullback-Leibler divergence		Between two Poisson laws with rates and

Properties and theorems

Property / theorem	Formula	Description
Binomial limit		Link to the binomial in the rare regime
Additivity	indep.	Intensities add over disjoint intervals
Thinning	, each event kept with prob.	Independent filtering of events; basis for jump models
Equidispersion		If empirically: overdispersion (negative binomial, etc.)
Normal approximation		For large ; see normal distribution
Link to the exponential		Number of arrivals by time in a Poisson process of rate
Random sum		Compound Poisson; aggregate loss models in insurance
Independent disjoint counts	—	Event counts on disjoint intervals are independent Poisson variables
Chernoff bound		Exponential upper tail bound

Applications

Arrivals and flows

Number of orders, trades, or network packets per minute/hour under constant rate and independence assumptions.

Defaults and claims

Number of defaults in a very granular portfolio or claims in non-life insurance (baseline frequency model).

Physics and reliability

Radioactive decays, rare equipment failures, production anomalies on a line.

Queues and microstructure

Elementary modeling of liquidity arrivals or ticks; extension to compound Poisson for price jumps.

Statistical inference

Concept	Formula	Description
Sample		independent intervals or observations
Sufficient statistic		summarizes all information on
MLE estimator		Sample mean; unbiased, consistent, efficient
Estimator variance		Decays as
Exact interval ()		Exact CI for from sum
Wald interval		Normal approximation; acceptable when is large
Exact test		Exact p-value under
Asymptotic test		Simple alternative for large
Conjugate prior (Bayesian)		The gamma law is conjugate to Poisson

Important remarks

Equidispersion () is a first diagnostic: a large gap suggests an alternative model.
Overdispersion is common in finance and insurance, motivating negative binomial or random-effects models.
The -based interval is preferable to Wald for small or moderate .
Comparing two rates and across samples amounts to testing equality of two Poisson means.

Computational methods

Method	Description	Applications
Pseudo-random generation	Inversion method (Knuth) or transformation from exponentials	Flow simulation, Monte Carlo
NumPy	`np.random.poisson(lam)` or `rng.poisson(lam, size=m)`	independent draws
SciPy	`scipy.stats.poisson(mu)` — PMF, CDF, `ppf`, `rvs`, moments	Exact probabilities, quantiles, simulation
Exact CI	`scipy.stats.chi2.ppf` via the gamma- relation	Confidence intervals for
Likelihood ratio test	Nested Poisson models (statsmodels `GLM` Poisson)	Poisson regression on covariates
Overdispersion diagnostic	Deviance / df; quasi-Poisson in `statsmodels`	Detecting poor Poisson fit

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References

• Wikipedia, Poisson distribution: https://en.wikipedia.org/wiki/Poisson_distribution

• See also Bernoulli, binomial, and stochastic processes.