Let Xᵢ, i = 1,..., n be independent continuous random variables, with Xᵢ having failure rate function rᵢ(t). Let T be independent of this sequence, and suppose that Σⁿᵢ₌₁ P{T = i} = 1.
Show that the failure rate function r(t) of Xᵀ is given by
ₙ
r(t) = ∑ rᵢ(t) P{T = i|X > t}
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