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.)