In actuarial science and demography, force of mortality, also known as death intensity,, that gives the instantaneous rate at which deaths occur at age x, conditional on survival to age x. In survival analysis it corresponds to the hazard function, and in reliability theory it corresponds to the failure rate. It has units of inverse time, and integrating it over an interval gives the survival probability over that interval.

Definition

be a non-negative random variable for its cumulative distribution function for its survival function.

. Formally, it is the limit of the conditional probability, divided by the interval length

is continuous with probability density function as

is differentiable, it is the negative derivative of the log-survival function

Interpretation and related quantities

is small enough that the rate does not change much over the interval.

is the hazard function. In reliability theory, the same mathematical object is commonly called the failure rate.

then the survival function can be expressed as

and, for a continuous lifetime distribution, the density can be written as

Survival probabilities and life tables

In actuarial notation, it is

Using the force of mortality, this conditional survival probability can be expressed as an exponential of the integrated force

. The force of mortality provides a continuous-age description that can be used to relate probabilities over different intervals through the integral relationship above.

Examples of mortality models

, corresponds to an exponential distribution and gives a memoryless survival pattern.

In actuarial work, the Gompertz–Makeham law of mortality., giving:

A common model in survival analysis and reliability uses a Weibull.

See also

• Failure rate

• Hazard function

• Actuarial present value

• Actuarial science

• Reliability theory

• Life expectancy

References