Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 61In closing this section, let us note that generalization to the case of manyrandom variables is again straightforward. The joint distribution function of nrandom variables X 1 ,X 2 ,...,X n ,or X, is given, by Equation (3.19), asF X …x† ˆP…X 1 x 1 \ X 2 x 2 ...\ X n x n †:…3:31†The corresponding joint density function, denoted by f X ( x), is thenf X …x† ˆqn F X …x†qx 1 qx 2 ...qx n;…3:32†if the indicated partial derivatives exist. Various properties possessed by thesefunctions can be readily inferred from those indicated for the two-randomvariablecase.3.4 CONDITIONAL DISTRIBUTION AND INDEPENDENCEThe important concepts of conditional probability and independence introducedin Sections 2.2 and 2.4 play equally important roles in the context ofrandom variables. The conditional distribution function of a random variable X,given that another random variable Y has taken a value y, is defined byF XY …xjy† ˆP…X xjY ˆ y†:…3:33†Similarly, when random variable X is discrete, the definition of conditional massfunction of X given Y ˆ y isp XY …xjy† ˆP…X ˆ xjY ˆ y†:…3:34†Using the definition of conditional probability given by Equation (2.24),we haveorp XY …xjy† ˆP…X ˆ xjY ˆ y†ˆ x \ Y ˆ y†ˆP…X;P…Y ˆ y†y†p XY …xjy† ˆpXY…x;; if pp Y …y† Y …y† 6ˆ 0; …3:35†TLFeBOOK