Two - Dimensional Random Variables So far we have considered only one dimensional RV i. e., we have considered The set of triples { x , y,, puj], where i = 1,2,3, ...,m, ... and j = 1, 2,3, ...,n, .. is such random experiments, the outcome of which had only one characteristic called the Joint Probability Distribution of (X,Y) . and hence was assigned a single real value. (2) If (X, Y) can assume all values in a specified region R in the xy - plane, In many situations, we will be interested in recording 2 or more characteristics (X, Y) is called a two - dimensional continuous RV. (numerically) of the outcome of a random experiment. Joint Probability Density Function of (X,Y) For example, If (X, Y) is a two - dimensional continuous RV such that. Both voltage and current might be of interest in a certain experiment. Definition : Let S be the sample space associated with a random experiment E. P x - dxxx+; dxandy-; dysY sy+; dy = f(x,y) dxdy, Let X = X(s) and Y = Y(s) be two functions each assigning a real number to each outcomes s E S. Then (X, Y) is called a two - dimensional RV. then f(x, y) is called the joint probability density function of (X, Y), provided Types of Two - dimensional Random Variables : f(x, y) satisfies the following conditions. (1) If the possible values of (X, Y) are finite or countably infinite, (X, Y) is called (i) f(x, y) 2 0, for all (x, y) ER, where R is the range space. a two - dimensional discrete RV. When (X, Y) is a two - dimensional discrete RV the possible values of (X, Y) 1) [ f(x,y ) dxdy = 1. may be represented as (x, y, ), i = 1,2, 3, ..., m, ...; j = 1,2,3, ..., n, ... Moreover if D is a subspace of the range space R, P{(X, Y) E D} is defined as Joint Probability Function of (X, Y) If (X, Y) is a two - dimensional discrete RV such that P{ (X, Y) ED) - | f(x,y) dxdy. D P(X = x Y = Vi) = Puj' then paj is called the joint probability mass function or simply the joint probability function of (X, Y) provided the following conditions are satisfied. (i) paj 2 0, for all i and j