Richard Bagby

Calculating normal probabilities

This is probably the best known example of an integral that cannot be evaluated in terms of elementary functions. In this note we develop an elementary approximation to P(a) which arises in a natural manner, is not difficult to use, and gives excellent results over the entire range 0 < a < oo. It is, in fact, more accurate than the four-place tables of values of P(a) commonly found in statistics textbooks, and simpler than many of the better known approximations (as described in Johnson and Kotz [1], for example). Our approximation to P(a) is

Covering lemmas and the sharp function

We present some new measure-theoretic inequalities for families of cubes covering open sets in RW and use these inequalities to estimate the Hardy-Littlewood maximal function Mf in terms of the sharp function f#.