(Solution) - Let F1 and F2 be d f s and let G be -(2025 Original AI-Free Solution)

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Let F1 and F2 be d.f.s, and let G be their convolution, G = F1 * F2. Then:
(i) If F1 is absolutely continuous with respect to Lebesgue measure with p.d.f. p1, it follows that G is absolutely continuous with respect to Lebesgue measure with p.d.f. p given by

(ii) Furthermore, if F2 is also absolutely continuous with respect to Lebesgue measure with p.d.f. p2, then