Fundamentals of Probability and Statistics for Engineers

148 Fundamentals of Probability and Statistics for Engineersf Y (y)1201 2 3yFigure 5. 21 Probability density function, f Y (y), in Example 5.17The problem is to obtain the joint probability distribution of random variablesY j , j ˆ 1,2,...,m, which arise as functions of n jointly distributed randomvariables X k , k ˆ 1,...,n. As before, we are primarily concerned with the casein which X 1 ,...,X n are continuous random variables.In order to develop pertinent formulae, the case of m ˆ n is first considered.We will see that the results obtained for this case encompass situations in whichm< n.Let X and Y be two n-dimensional random vectors with components(X 1 ,...,X n ) and (Y 1 ,...,Y n ), respectively. A vector equation representingEquation (5.60) isYˆg…X†…5:61†where vector g( X) has as components g 1 (X), g 2 ( X),...g n (X). We first considerthe case in which functions g j in g are continuous with respect to each of theirarguments, have continuous partial derivatives, and define one-to-onemappings. It then follows that inverse functions g j1 of g 1 , defined byXˆg 1 …Y†;…5:62†exist and are unique. They also have continuous partial derivatives.In order to determine f Y ( y) in terms of f X (x), we observe that, if a closedregion R n X in the range space of X is mapped into a closed region Rn Y in therange space of Y under transformation g, the conservation of probability givesZZZf Y …y†dy ˆZf X …x†dx;…5:63†R n YR n XTLFeBOOK