# Best approximation of binomial distribution

I want to approximate the distribution of a binomial random variable $X\sim B(n,p)$. In all references I have seen it is done by a normal distribution. (Of course the Central Limit Thm and de Moivre–Laplace thm are good justifications for it.) Doesnt though the truncated normal distribution on [0,n] give a better approximation? (It eliminates negative values). Or perhaps a Beta distribution?

Are there any books or papers where "goodness" of these approximations is analyzed? (I know that there is a Bayesian point of view on this, but please go easy with the technical details on that with me as I am not familiar with that theory.)

• To answer the question one has to establish a quantitative understanding of "better." Some common meanings are (1) in terms of average error in approximating individual Binomial probabilities, (2) errors in approximating tail probabilities, (3) errors in approximating central probabilities. All these errors can be assessed in different ways--as absolute differences, relative differences, with various weights, and so on. Then the scope of $n$ and $p$ must be specified: what will their typical values be? In most cases I believe the truncated Normal will actually be worse. – whuber Jun 24 '14 at 21:24

A truncated Normal distribution might also be fitted to the Binomial distribution for approximation, but it then cannot be used interchangeably with the parameters $\mu=np, \sigma^2=np(1-p)$ because it would have a different skew than before; in particular, the mean would be shifted to the right.
The non-truncated Normal distribution with parameters $N(np, np(1-p))$ has the nice property, that its mean and variance are exactly equal to that of a Binomial distribution, so it is a sort of Method of Generalized Moments best-fit.
The theoretic justification of Normal distribution for Binomial, is that they converge together for large $n$: https://www.youtube.com/watch?v=u9onO78hDlw So for large $n$, the mean shifts more and more to the right such that the negative value issue becomes more and more irrelevant while the density shapes converge.