If *X* ~ Poisson(λ) with λ “large” then *X* is well approximated by a normal distribution. The approximation that falls out of the central limit theorem approximates the CDF of *X* by

*F _{X}*(

*k*) ≈ Φ((

*k*+ 0.5 − λ)/√ λ).

Here Φ is the CDF of a standard normal (Gaussian) random variable. The central limit theorem approximation is studied in these notes. The **Wilson-Hilferty approximation** improves on the classical approximation by using a non-linear transformation of the argument k. This approximation uses

*F _{X}*(

*k*) ≈ Φ((

*c*− μ)/σ)

where

*c* = (λ/(1 + *k*))^{1/3}, μ = 1 − 1/(9*k* + 9), and σ = 1/(3 √(1 + *k*)).

## Example

The graph below gives the error for the normal approximation to the CDF of a Poission(10) random variable using *F _{X}*(

*k*) ≈ Φ((

*k*+ 0.5 − λ)/√ λ).

Here is the corresponding graph using the Wilson-Hilferty approximation.

The maximum error in the classical approximation is 0.0207. The maximum error in the W-H approximation is 0.00049, about 42 times smaller. This is typical: the error is often a couple orders of magnitude smaller in the W-H approximation than the classical approximation.

For more information, see “Some Suggestions for Teaching About Normal Approximation to Poisson and Binomial Distribution Functions” by Scott M. Lesch and Daniel R. Jeske, *The American Statistician*, August 2009, Vol 63, No 3.

See also notes on the normal approximation to the beta, binomial, gamma, and student-t distributions.