(Solution) - In Example 5 2 10 a partial fraction decomposition is needed to -(2025 Original AI-Free Solution)

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In Example 5.2.10, a partial fraction decomposition is needed to derive the distribution of the sum of two independent Cauchy random variables. This exercise provides the details that are skipped in that example.
(a) Find the constants A, B, C, and D that satisfy

where A, B, C, and D may depend on z but not on w.
(b) Using the facts that
evaluate (5.2.4) and hence verify (5.2.5).
(That the integration in part (b) is quite delicate. Since the mean of a Cauchy does not exist, the integrals
dw and
dw do not exist.
However, the integral of the difference does exist, which is all that is needed.)