The hazard function, conventionally denoted 
, is defined as the event rate at time t conditional on survival until time t or later (that is, T ≥ t),
, is defined as the event rate at time t conditional on survival until time t or later (that is, T ≥ t),
Force of mortality is a synonym of hazard function which is used particularly in demography and actuarial science, where it is denoted by 
. The term hazard rate is another synonym.
. The term hazard rate is another synonym.The hazard function must be non-negative, λ(t) ≥ 0, and its integral over ![[0, \infty]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u1r8Fnm9bftoYU8SZPBl2SrGHgerO7odS6klXcJyNBqTYZ6WiFUPMuwwjR_-POlRHK1lQEzCFDneXY-UUXDsinLo7vzIK1XT23uCpM8ZBRwJ3BAmxWgyDDkCFkqr_hRROy77ZSYkosNsvQ5Eh4ubDsTgH_t_HVlX_IPr4=s0-d)
must be infinite, but is not otherwise constrained; it may be increasing or decreasing, non-monotonic, or discontinuous. An example is the bathtub curve hazard function, which is large for small values of t, decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.
must be infinite, but is not otherwise constrained; it may be increasing or decreasing, non-monotonic, or discontinuous. An example is the bathtub curve hazard function, which is large for small values of t, decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.The hazard function can alternatively be represented in terms of the cumulative hazard function, conventionally denoted 
:
:
so transposing signs and exponentiating
or differentiating (with the chain rule)
The name "cumulative hazard function" is derived from the fact that
which is the "accumulation" of the hazard over time.
From the definition of 
, we see that it increases without bound as t tends to infinity (assuming that S(t) tends to zero). This implies that 
must not decrease too quickly, since, by definition, the cumulative hazard has to diverge. For example, 
is not the hazard function of any survival distribution, because its integral converges to 1.
, we see that it increases without bound as t tends to infinity (assuming that S(t) tends to zero). This implies that must not decrease too quickly, since, by definition, the cumulative hazard has to diverge. For example, is not the hazard function of any survival distribution, because its integral converges to 1.
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