Normal Approximations and the Central Limit Theorem

Abstract

Many of the special discrete and special continuous distributions that we have discussed can be well approximated by a normal distribution for suitable configurations of their underlying parameters. Typically, the normal approximation works well when the parameter values are such that the skewness of the distribution is small. For example, binomial distributions are well approximated by a normal distribution when n is large and p is not too small or too large. Gamma distributions are well approximated by a normal distribution when the shape parameter α is large. Whenever we see a certain phenomenon empirically all too often, we might expect that there is a unifyingmathematical result there, and in this case indeed there is. The unifyingmathematical result is one of the most important results in all of mathematics and is called the central limit theorem. The subject of central limit theorems is incredibly diverse. In this chapter, we present the basic or the canonical central limit theorem and its applications to certain problems with which we are already familiar. Among numerous excellent references on central limit theorems, we recommend Feller (1968, 1971) and Pitman (1992) for lucid expositions and examples. The subject of central limit theorems also has a really interesting history; we recommend Le Cam (1986) and Stigler (1986) in this area. Careful and comprehensive mathematical treatments are available in Hall (1992) and Bhattacharya and Rao (1986). For a diverse selection of examples, see DasGupta (2008).