# 1-Dimensional Cumulative Distribution Functions and Bounded Variation Functions

• Richard M. Meyer
Part of the Universitext book series (UTX)

## Abstract

A real-valued function F(x), defined for all real x, is termed a 1-dimensional cumulative distribution function (c.d.f.) iff it satisfies the following four basic properties:
$$\begin{array}{*{20}{c}} {(i){{x}_{1}} \leqslant {{x}_{2}} \Rightarrow F({{x}_{1}}) \leqslant F({{x}_{2}})[non - decrea\sin g]} \\ {(ii)F(x) \to 1asx \to + \infty [F( + \infty ) = 1]} \\ {(iii)F(x) \to 0asx \to - \infty {{{[F( - \infty ) = 0]}}^{1}}} \\ {(iv)F(x - \varepsilon ) \to F(x)as\varepsilon \downarrow 0[left - continuity]} \\ \end{array}$$
Cumulative distribution functions are fundamental, and our concern here is with certain Mathematical properties of c.d.f.’s that will be useful later when studying Bounded Variation Functions and the Riemann-Stieltjes Integral.

## Keywords

Cumulative Distribution Function Bounded Interval Unbounded Interval Mathematical Property Pointwise Convergence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References to Additional and Related Material: Section 8

1. 1.
Feller, W., “An Introduction to Probability Theory and its Applications”, Vol. II, Wiley and Sons, New York (1966).
2. 2.
Hobson, E., “Theory of Functions of a Real Variable”, Vol. I, Dover Publications, Inc., New York.Google Scholar
3. 3.
McShane, E. and T. Botts, “Real Analysis”, VanNostrand, Inc., New York (1959).
4. 4.
Royden, H., “Real Analysis”, Macmillan Co., New York (1964).Google Scholar
5. 5.
Simmons, G., “Topology and Modern Analysis”, McGraw-Hill, Inc., New York (1963).
6. 6.
Titchmarsh, E., “The Theory of Functions”, (Second Ed.), Oxford University Press, (1939).Google Scholar

© Springer-Verlag New York Inc. 1979

## Authors and Affiliations

• Richard M. Meyer
• 1
1. 1.Niagara University, College of Arts and SciencesNiagara UniversityUSA