The volume cut off a simplex by a half-space

A formula for the volume cut off an w-dimensional simplex by a half-space has immediate application in probability theory.This note presents a derivation of such a formula in a short and completely elementary way and also yields the moments of this volume about the coordinate hyperplanes, and the volumes cut off the lower dimensional faces of the simplex and their moments.Before stating our results we introduce some notation.The "minus function" (x)t equals x k if x < 0 and is zero otherwise.The nth divided difference of a real-valued function fix) is the symmetric function of n + 1 arguments defined inductively byIf x 0 = x n we define D{f(x): x 0 , x l9 , a? Λ _!, α; 0 } = J-D{f(x): x 0 , x lf , a? n _J