The characteristic function of probability theory

Includes bibliographical references.This paper is concerned with a particularly useful function of probability theory. The function is the characteristic function of a random variable. In this paper its usefulness and importance is demonstrated by way of three significant theorems. Firstly, the one-to-one correspondence between the characteristic function and the distribution function of a random variable is shown. Secondly, this one-to-one correspondence is shown to be continuous in the sense that a sequence of distribution functions converges to a distribution function F(•) if and only if the corresponding sequence of characteristic functions converges to the characteristic function ϕ(•) of F(•). Thirdly, the real power of the characteristic function is demonstrated when the central limit theorem is proved, which is the most important theorem in probability theory.M.S. (Master of Science