Poisson Approximation to Binomial
When n is large and p is small, the binomial Bin(n,p) is very close to the Poisson with rate lambda = np. This is the right approximation for rare-event claim counts where the normal approximation would be miscalibrated, and the Le Cam bound says exactly how close.
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Plain-English Definition
If you have a large number n of independent trials, each with a small success probability p, the number of successes is well approximated by a Poisson distribution with rate lambda equal to n times p. The approximation gets better as p shrinks and n grows, with the expected count np held fixed.
This is the rare-event regime: many chances, each one unlikely. It shows up in catastrophe modeling, individual-claim counts on low-frequency coverages, large-deductible exposures, and any setting where a small probability is multiplied by a large exposure.
Worked Example
A book of 5000 small-business policies has independent annual probability 0.004 of triggering a particular liability claim. The expected number of claims per year is lambda = 5000 times 0.004 = 20. To approximate the probability of seeing at most 15 claims in a year, use Pois(20) and read off the CDF.
From the Poisson CDF, P(X ≤ 15) when lambda = 20 is about 0.157. The Le Cam total-variation bound says the true binomial probability is within n times p squared = 5000 times 0.000016 = 0.08 of this value. The approximation is well inside the tolerance you need for capital or reinsurance pricing.
Why Actuaries Use It
The Poisson approximation is what makes rare-event claim-count modeling tractable. Whenever you see lambda quoted as a per-period claim rate without an underlying binomial trial count, the implicit move is the Poisson approximation.
It is also the entry point to compound Poisson models, the standard framework for aggregate-loss distributions in FAM and ASTAM. The frequency component there is Poisson precisely because the underlying portfolio is many independent low-probability exposures.
When To Use Poisson Versus Normal
Two approximations to the binomial coexist on Exam P. The Poisson is sharp when p is small and np is moderate. The normal approximation, by way of the De Moivre-Laplace theorem, is sharp when np and n times (1 minus p) are both at least 10.
A useful rule of thumb. Use Poisson when p is at most 0.05 and n is at least 100. Use the normal (with continuity correction) when np is at least 10 and n times (1 minus p) is at least 10. Both regimes can overlap; when in doubt, the Le Cam bound and the Berry-Esseen bound tell you which approximation has lower error at the (n, p) you actually care about.
Common Mistakes
Trying to use the normal approximation when p is tiny gives a symmetric approximation to a strongly right-skewed distribution. At lambda = 2, the Poisson skewness is about 0.71, far from Gaussian. The normal-approximation answer to P(X = 0) or P(X = 1) will be wrong by a non-trivial amount for small lambda.
Confusing the Poisson distribution with the Poisson process. The distribution is the count of rare events in one fixed window. The process is the random arrangement of arrival times along the real line, with the count on any window being Poisson. The approximation here is for the distribution.
Statistics Connection
Maximum likelihood estimation of a Poisson rate from observed counts is the sample mean. This is the same estimator the law of large numbers justifies as consistent. Together, the Poisson approximation and the LLN explain why empirically estimating a claim rate from a year of policy data and treating it as a Poisson parameter is internally consistent.