Partner with ConvertIt.com
 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | MEDIUM | LARGE Below is the OCR-scanned text from this page: 26. Probability Functions Mathematical Properties 26.1. Probability Functions: Definitions and Properties Univariate Cumulative Distribution Functions A real-valued function F(x) is termed a (uni- variate) cumulative distribution function (c.d.f.) or simply distribution function if i) F(x) is non-decreasing, i.e., F ( z l ) l F ( r z ) for ii) F(x) is everywhere continuous from the iii) F(--)=O, F(-)=1. The function F(x) signifies the probability of the event “XO is termed the domain of the random variable X . A discrete distribution of a random variabIe is called a lattice distribution if therL exist numbers a and b #O such that every possible value of X can be represented in the form a+bn where n takes on only integral values. A summary of some properties of certain discrete distributions is presented in 26.1.19-26.1.24. Continuous Distributions. Continuous distri- butions are characterized by F(x) being absolutely continuous. Hence F(x) possesses a derivative F’(x)=-f(z) and the c.d.f. can be written 26.1.2 F(x)=Pr{XO make up the domain of the random variable X . A summary of some properties of certain selected continuous distributions is presented in 26.1.25-26.1.34. Multivariate Probability Functions The real-valued function F(xl, xz, . . . 2,) defines an n-variate cumulative distribution func- tion if i) F(zl, xz, . . . 2,) is a non-decreasing func- tion for each xi ii) F(xl, x2, . . . z,) is continuous from the right in each xi; i.e., F ( s , x2, . . . 2,) =lim F ( q , . . ., xi+€, . . ., 2,) F ( - , a,. . ., -)=1. iv) F(xl, z2, . . ., 2,) assigns nonnegative prob- ability to the event xlO for k=l, 2, . . ., n. d+ * iii) F(xl, zz, . . . t,) = O when any xi= - Q) ; xz