Renewal Theory & Hazard Rate

Renewal processes with different hazard rate functions. The hazard h(t) = f(t)/S(t) governs how failure risk changes over time — shaping survival curves, inter-renewal distributions, and long-run rates.

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Hazard h(t)
Survival S(t)
PDF f(t)
Renewal events
Survival: S(t) = exp(-∫₀ᵗ h(u) du)  |  PDF: f(t) = h(t)·S(t)  |  Mean inter-renewal: μ = ∫₀∞ S(t) dt