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Define $X$ ~ Pareto($a$) and $Y$ ~ Pareto($b$), meaning $f_X (x) = ax^{-a + 1}$ for $x \geq 1$ and $f_Y (y) = by^{-b + 1}$ for $y \geq 1$.

Assuming that $X$ and $Y$ are independent random variables, how would I find the density of $Z = X + Y$ and $W = Z - Y$.

Franck Dernoncourt
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Kevin
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    Are you after the joint density of Z and W or their marginal densities? (What are you using it for?) For $Z$, take a look [here](http://stats.stackexchange.com/questions/115291/what-distribution-results-in-adding-two-pareto-distributions). – Glen_b Oct 16 '16 at 02:22
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    The marginal distribution of $W = Z-Y = (X+Y)-Y = X$ is particularly easy to find. :-) – Dilip Sarwate Oct 16 '16 at 02:34
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    The density given $f_X (x) = ax^{-a + 1}$ is not Pareto (nor likely a valid density). Pareto for $x>1$ would be $f(x) = ax^{-(a+1)}$ – wolfies Oct 16 '16 at 06:16

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