The hazard function h(x) in terms of the density f(x) and the survival function S(x) is given by which expression?

• A. d/dx [ln S(x)]
• B. S(x)/f(x)
• C. ∫_0^x f(t) dt
• D. f(x)/S(x)

Answer: (D)

The hazard function h(x) is the instantaneous failure rate at age x given survival to x. For a continuous variable with density f and survival S, it equals h(x) = f(x) / S(x). This comes from conditioning on having survived to x and looking at the instantaneous chance of failure in a tiny interval around x. Equivalently, h(x) = -d/dx [ln S(x)], since d/dx ln S(x) = S'(x)/S(x) = -f(x)/S(x). The other expressions do not give the hazard: ∫_0^x f(t) dt equals the CDF F(x); and S(x)/f(x) is the reciprocal of the hazard.