In this section, we consider transformations from two independent random variables to a single random variable. Except for the sum of random variables described first, we provide derivations only for the case when both random variables are continuous; the results for discrete random variables, or when one is continuous and the other is discrete (as in Example 4.15) can be determined from of the continuous results.

4.9.1 Sum: Z = X+Y

The cdf (indirect) method will be used to derive the pdf of Z when X and Y are independent continuous random variables. Starting with the joint cdf fX,Y(x, y), the cdf of Z is written as the following double integral:

(4.148) Numbered Display Equation

where in the last expression we have rewritten as and used it to limit the range of the inner integral. Differentiating with respect to z gives

(4.149) Numbered Display Equation

This is a general result for dependent X and Y. When they are independent, the joint pdf splits, giving the final result

(4.150) Numbered Display Equation

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