Force of mortality

In actuarial science and demography, force of mortality, also known as death intensity, is a function, usually written

μ

(

x

)

{\displaystyle \mu (x)}

, 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

Let

X

{\displaystyle X}

be a non-negative random variable representing an individual's age at death (or lifetime). Write

F

(

x

)

=

Pr

(

X

≤

x

)

{\displaystyle F(x)=\Pr(X\leq x)}

for its cumulative distribution function and

S

(

x

)

=

Pr

(

X

>

x

)

{\displaystyle S(x)=\Pr(X>x)}

for its survival function.

The force of mortality at age

x

{\displaystyle x}

, written

μ

(

x

)

{\displaystyle \mu (x)}

, is defined as the instantaneous conditional rate of death at age

x

{\displaystyle x}

. Formally, it is the limit of the conditional probability of dying in a short interval after

x

{\displaystyle x}

, divided by the interval length:

μ

(

x

)

=

lim

Δ

x

→

0

+

Pr

(

x

<

X

≤

x

+

Δ

x

∣

X

>

x

)

Δ

x

.

{\displaystyle \mu (x)=\lim _{\Delta x\to 0^{+}}{\frac {\Pr(x<X\leq x+\Delta x\mid X>x)}{\Delta x}}.}

When

X

{\displaystyle X}

is continuous with probability density function

f

(

x

)

{\displaystyle f(x)}

, the force of mortality can be written in terms of

f

{\displaystyle f}

and

S

{\displaystyle S}

as

μ

(

x

)

=

f

(

x

)

S

(

x

)

=

f

(

x

)

1

−

F

(

x

)

.

{\displaystyle \mu (x)={\frac {f(x)}{S(x)}}={\frac {f(x)}{1-F(x)}}.}

Equivalently, where

S

{\displaystyle S}

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

μ

(

x

)

=

−

d

d

x

ln

⁡

S

(

x

)

.

{\displaystyle \mu (x)=-{\frac {\mathrm {d} }{\mathrm {d} x}}\ln S(x).}

Interpretation and related quantities

The force of mortality

μ

(

x

)

{\displaystyle \mu (x)}

is an instantaneous rate rather than a probability.