This page is an appendix to the page Error in the normal approximation to the Poisson distribution.

The previous page noted that a Poisson random variable with mean λ has the same distribution as the sum of *N* independent Poisson random variables *X _{i}* each with mean λ/

*N*. The Berry-Esséen theorem says that the error in approximating the CDF of ∑

*X*with the CDF of its normal approximation is uniformly bounded by

_{i}*C*ρ/σ

^{3}√

*N*where

*C*is a constant we discuss below, ρ = E(|

*X*– λ/

_{i}*N*|

^{3}) and σ, the standard deviation of

*X*, equals (λ/

_{i}*N*)

^{1/2}.

Since the number *N* is arbitrary, we pick it as large as we like. This paper says the constant in the Berry-Esséen theorem can be made less than 0.7164 if *N* ≥ 65. This is no problem here because we will be taking the limit as *N* goes to infinity.

Next we calculate ρ, the third absolute moment of *X _{i}*. Let ω = λ/

*N*. We can get an upper bound on ρ as follows.

From this we can conclude that ρ/σ^{3}√*N* equals λ plus term involving 1/*N*. As we let *N* to infinity, the latter terms drop out and so we have the error bound on the normal approximation to the Poisson of *C*/√λ where *C* < 0.7164.