(Solution) - Suppose that X and Y are independent random variables Suppose -(2025 Original AI-Free Solution)

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Suppose that X and Y are independent random variables. Suppose that X has a discrete distribution concentrated on finitely many distinct values with p.f. f1. Suppose that Y has a continuous distribution with p.d.f. f2. Let Z = X + Y. Show that Z has a continuous distribution and find its p.d.f. First find the conditional p.d.f. of Z given X = x.