In Example 5.2.10, a partial fraction decomposition is needed to derive the distribution of the sum of

Question:

In Example 5.2.10, a partial fraction decomposition is needed to derive the 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

$In Example 5.2.10, a partial fraction decomposition is needed to$

where A, B, C, and D may depend on z but not on w.
(b) Using the facts that

$In Example 5.2.10, a partial fraction decomposition is needed to$

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

$In Example 5.2.10, a partial fraction decomposition is needed to$

dw and

$In Example 5.2.10, a partial fraction decomposition is needed to$

dw do not exist.
However, the integral of the difference does exist, which is all that is needed.)

Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...