Let X, Y have the joint probability density f (x, y). Then the integral h(z) =

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Let X, Y have the joint probability density f (x, y). Then the integral h(z) = ∞

−∞ f (y − z, y)dy is finite for almost all z, and is the probability density of Z = Y − X.

[Since P{Z ≤ b} = b

−∞ h(z)dz, it is finite and hence h is finite almost everywhere.]

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