#### Publication Date

1966

#### Document Type

Dissertation/Thesis

#### First Advisor

Carlborg, Frank W. (Frank William), 1928-||Christiano, John G.

#### Degree Name

M.S. (Master of Science)

#### Legacy Department

Department of Mathematics

#### LCSH

Distribution (Probability theory); Probabilities

#### Abstract

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.

#### Recommended Citation

Foley, David Michael, "The characteristic function of probability theory" (1966). *Graduate Research Theses & Dissertations*. 5353.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/5353

#### Extent

v, 26 pages

#### Language

eng

#### Publisher

Northern Illinois University

#### Rights Statement

In Copyright

#### Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

#### Media Type

Text

## Comments

Includes bibliographical references.