Poisson Distribution

Where the formula comes from

Take a stretch of time of length T. Suppose events fall on it independently at an average rate of λ events per unit time, and the chance of two events landing on the same instant is zero. We want the probability that exactly k events fall inside our window. The Poisson answer is

P(X = k) = λ^k · e^(-λ) / k!     for k = 0, 1, 2, ...

(Here λ is taken to be the average count over the window — if the rate is r per second and the window is 5 seconds, then λ = 5r.) The shape of the PMF is governed by a single positive number. For λ < 1 the most likely count is 0 and the distribution is sharply right-skewed. For λ around 1 the curve has a flat hump near 0 and 1. For larger λ it rounds out into something that looks more and more like a bell curve, and in fact Poisson(λ) is well-approximated by Normal(λ, λ) once λ exceeds about 20.

The cleanest derivation runs through the Binomial. Slice the time window into n equal sub-intervals of length T/n. Make n large enough that the probability of two events in a single sub-interval is negligible. Let p be the chance an event lands in any given sub-interval. Then the number of sub-intervals containing an event is Binomial(n, p). To keep the average count λ = np constant as we refine the slicing, we send n → ∞ and p → 0 with np = λ fixed. The Binomial PMF

C(n, k) p^k (1-p)^(n-k)

tends, term by term, to λ^k e^(-λ)/k! by Stirling and the limit (1 − λ/n)^n → e^(-λ). The Poisson is what the Binomial relaxes to when trials are continuous and successes are rare.

Mean equals variance, and other one-parameter consequences

The Poisson is a one-parameter family — λ controls everything. Two consequences are unusually useful in practice:

1. Mean = Variance = λ. A direct sum gives E[X] = λ, and a similar manipulation on E[X(X−1)] yields Var[X] = λ. So if you fit a Poisson to data, you only need one moment to estimate the parameter, and you can sanity-check the fit by comparing the sample mean to the sample variance — they should be close.
2. Additivity. X ~ Poisson(λ₁), Y ~ Poisson(λ₂), independent ⇒ X + Y ~ Poisson(λ₁ + λ₂). This is why aggregated streams of independent events stay Poisson: thousand users with their own arrival rates sum into a Poisson process at the merged rate.

The moment generating function is M(t) = exp(λ(e^t − 1)). The skewness is 1/√λ — heavily right-skewed for small λ, nearly symmetric for λ in the hundreds. The kurtosis excess is 1/λ — more peaked than normal at small λ, normal-like at large λ.

From distribution to process

The Poisson distribution describes counts in a single window. The Poisson process glues these together into a continuous-time arrival model. Three equivalent definitions all pick out the same object:

• The number of events in any window of length t is Poisson(λt), and counts in disjoint windows are independent.
• The probability of exactly one event in a small window of length h is λh + o(h); the probability of two or more is o(h); and the count is independent of past arrivals.
• The waiting times between consecutive events are i.i.d. Exponential(λ).

The third characterisation is the one that makes simulation easy. To generate a Poisson process at rate λ, draw exponential gaps and accumulate them — the kth event time is the sum of k Exp(λ) variables, which has a Gamma(k, λ) distribution. The Poisson is the marginal count law of this clock.

Memorylessness is the philosophical point. If you've been waiting 30 seconds for the next bus and the gaps are Exp(λ), the distribution of your remaining wait is still Exp(λ) — the past wait carries no information about the future. Real bus schedules don't have this property, but radioactive nuclei and TCP packet arrivals do.

Worked example: ER admissions

An emergency department records on average 3.5 admissions per hour. Assume admissions arrive as a Poisson process. What is the probability that exactly 5 patients arrive in a given hour?

λ = 3.5
k = 5
P(X = 5) = (3.5)^5 · e^(-3.5) / 5!
        = 525.219 · 0.030197 / 120
        ≈ 0.1322

So about a 13% chance. The full PMF rounds out as follows for the same hour:

P(X = 0) ≈ 0.0302
P(X = 1) ≈ 0.1057
P(X = 2) ≈ 0.1850
P(X = 3) ≈ 0.2158       ← mode
P(X = 4) ≈ 0.1888
P(X = 5) ≈ 0.1322
P(X = 6) ≈ 0.0771
P(X ≥ 7) ≈ 0.0653

From this you can answer staffing questions. The 95th percentile is 7 admissions; if you want to be 95% sure of having capacity, plan for 7. The probability of more than 10 admissions is about 0.4% — a tail event but not negligible across a year of hours.

Now suppose the night shift sees the same rate over an 8-hour period. The total count is Poisson(28), and by the additivity property you can reach this either by summing eight independent hourly Poisson(3.5) or by viewing the 8-hour window directly. Both perspectives are mathematically the same, which is the entire point of the Poisson process.

Poisson, Binomial, and Normal at a glance

	Binomial(n, p)	Poisson(λ)	Normal(μ, σ²)
Support	{0, 1, ..., n}	{0, 1, 2, ...}	(−∞, ∞)
Mean	np	λ	μ
Variance	np(1−p)	λ	σ²
Parameters	2 (n, p)	1 (λ)	2 (μ, σ²)
Discrete?	Yes	Yes	No
Use when	Fixed n trials	Rare events, fixed window	Continuous, large samples
Limiting case of	—	Binomial as n→∞, p→0	Both, by CLT
Sum of independents	Same family iff equal p	Always Poisson	Always Normal

The three distributions sit on a single ladder. Binomial(n, p) describes a fixed batch of yes/no trials. Push n high and p low so that np = λ is fixed and you get Poisson(λ). Push λ high and you get Normal(λ, λ) by the central limit theorem. The right tool depends on which limits dominate your data.

Where the Poisson distribution shows up

• Radioactive decay. A gram of radium emits about 3.7×10¹⁰ alpha particles per second. The count in any second-long window is Poisson with that mean. Geiger-counter electronics are calibrated against the predicted Poisson PMF — anomalies in the variance indicate detector dead-time or pile-up.
• Premier League goals. Match-by-match goal counts fit Poisson with λ ≈ 1.35 per team per match. Bookmakers price over/under markets using the implied Poisson — and the over-2.5-goals line of about 50% probability falls naturally out of two independent Poisson(1.35) team rates summed.
• Bortkiewicz's Prussian cavalry data. 1875–1894 horse-kick deaths in 14 corps. Total 196 deaths over 280 corps-years, λ ≈ 0.7. Predicted vs observed counts of corps-years with 0/1/2/3+ deaths: 144/91/32/13 expected, 144/91/32/13 observed. The fit is so good it became the Poisson's founding case study.
• Web server requests. Aggregate request streams to a busy web server are typically modelled as Poisson once you condition on time-of-day. Capacity planning uses Erlang-C (a queueing formula) which assumes Poisson arrivals into a server pool, and the SLA tail is computed from the Poisson PMF.
• Mutation counts in genome sequencing. Per-base mutation counts in DNA sequencing reads at low coverage are Poisson with λ equal to the mean coverage. Bayesian variant callers like GATK use the Poisson likelihood as their per-site model.

Estimating λ from data

The maximum likelihood estimator of λ from a sample x₁, ..., x_n is just the sample mean:

λ̂_MLE = (x₁ + x₂ + ... + x_n) / n

The MLE is unbiased, efficient, and equal to the method-of-moments estimator. The (asymptotic) standard error is √(λ̂/n), so a 95% confidence interval is approximately λ̂ ± 1.96 √(λ̂/n). For small counts, exact confidence intervals based on the chi-squared distribution are preferred:

Lower:  ½ χ²(α/2; 2k)         where k = total count
Upper:  ½ χ²(1 − α/2; 2k+2)

Goodness-of-fit is checked with a chi-squared test bucketing observed counts vs Poisson-predicted counts, or with the dispersion test that compares sample variance to sample mean. A dispersion ratio (variance/mean) much larger than 1 is a red flag — switch to Negative Binomial.

Variants and extensions

• Compound Poisson. Sum of N i.i.d. random jumps where N is Poisson(λ). Used in insurance for total claims (number of claims is Poisson, claim sizes are heavy-tailed) and in finance for jump-diffusion models like Merton 1976.
• Inhomogeneous Poisson process. The rate λ(t) varies in time. Counts in any window are Poisson with mean ∫λ(t) dt over the window. Used in seismology, where aftershock rates follow Omori's law λ(t) ∝ 1/(t + c).
• Spatial Poisson process. The same idea in 2D or 3D — count of points in any region A is Poisson(∫_A λ(x) dx). Used in cosmology for galaxy distributions, in forestry for tree placement, in cellular biology for receptor positions on a membrane.
• Zero-inflated Poisson. A mixture: with probability p the count is forced to 0, otherwise it is Poisson(λ). Models data with an excess of zeros — visits to a clinic, accidents at safe intersections.
• Negative Binomial. Poisson-Gamma mixture: λ itself is drawn from a Gamma distribution per observation. Allows variance > mean (overdispersion) and is the standard fix when raw Poisson regression fails its dispersion test.

Common pitfalls

• Treating dependent events as independent. Goal scoring after a red card is not independent of the red card. Fitting a single λ across heterogeneous match contexts produces biased predictions even if the marginal mean looks right.
• Ignoring overdispersion. If the sample variance is twice the sample mean, Poisson confidence intervals are off by √2. The model fits the mean but lies about the spread, and downstream inferences (p-values, prediction intervals) are correspondingly wrong.
• Forgetting the rate vs count distinction. λ is not an instantaneous probability — it is an expected count in the window you have specified. Doubling the window doubles λ. Saying "λ = 0.7 per year" is meaningful; saying "λ = 0.7" with no window is not.
• Using Poisson when n is fixed and known. If you are doing 100 coin flips, use Binomial(100, p). The Poisson approximation is for the regime where n is enormous and p is small enough that n itself becomes irrelevant. With small n the Poisson tail at counts > n is positive but the Binomial cuts cleanly.
• Reading the PMF as a probability density. The Poisson is discrete. P(X = 1.5) is not 0.07 from some interpolated curve — it is undefined. Plot bars, not lines, and compare to integer bins of the data.