Which expression correctly defines the cumulative hazard function H(x) in terms of the hazard rate h(t)?

• A. H(x) = ∫_0^x h(t) dt
• B. H(x) = S(x)
• C. H(x) = -∫_0^x h(t) dt
• D. H(x) = ∫_0^x f(t) dt

Answer: (A)

Hazard rate h(t) is the instantaneous risk of failure at time t for someone who has survived to t. To accumulate that risk up to time x, you integrate h(t) over time from 0 to x. That sum of instantaneous risks, ∫_0^x h(t) dt, defines the cumulative hazard H(x). A key consequence is the relation to the survival function: S(x) = exp(-H(x)). The other forms don’t fit because S(x) is a survival probability, not a hazard accumulation; a negative integral would have no physical meaning for hazard; and integrating f(t) would give the distribution function F(x), not the cumulative hazard.