Which expression correctly describes the hazard function in terms of density and survival?

• A. h(t) = S(t)/f(t).
• B. h(t) = F(t)/S(t).
• C. h(t) = f(t)/S(t).
• D. h(t) = 1 - F(t).

Answer: (C)

Hazard at time t is the instantaneous risk of failure given that the subject has survived up to t. It is defined as the conditional probability of failing in a very small interval around t, divided by the length of that interval: h(t) = lim_{dt→0} P(t ≤ T < t+dt | T ≥ t) / dt. Since P(t ≤ T < t+dt) ≈ f(t) dt and P(T ≥ t) = S(t), this becomes h(t) = f(t) / S(t). In words, the hazard is the density of the event time scaled by the probability of having survived to that time. This differs from the survival function itself (S(t)) or the cumulative distribution (F(t)); using those directly does not yield the instantaneous failure rate. The density and survival relationship also ties to the cumulative hazard via S(t) = exp(-H(t)) and f(t) = h(t) S(t).