PROBABILITY THEORY:EXPECTATIONS FOR BIVARIATE DISTRIBUTIONS

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EXPECTATIONS FOR BIVARIATE DISTRIBUTIONS

Section 24.7 defined the expectation of a function of a univariate random variable. The expectation of a function of a bivariate random variable (X1, X2) may be defined in a sim- ilar manner. Let g(X1, X2) be a function of the bivariate random variable (X1, X2). Let

INTRODUCTION TO OPERATIONS RESEARCH-0585

An alternate definition can be obtained by recognizing that Z = g(X1, X2) is itself a univariate random variable and hence has a density function if Z is continuous and a probability distribution if Z is discrete. The expectation of Z for these cases has already been defined in Sec. 24.7. Of particular interest here is the extension of the theorem of the un- conscious statistician, which states that if (X1, X2) is a continuous random variable and if Z has a density function hZ(y), then

INTRODUCTION TO OPERATIONS RESEARCH-0586

INTRODUCTION TO OPERATIONS RESEARCH-0587

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