In: Statistics and Probability
::::Answer:::
In actuarial science, force of mortality represents the instantaneous rate of mortality at a certain age measured on an annualized basis. It is identical in concept to failure rate, also called hazard function, in reliability theory.
In a life table, we consider the probability of a person dying from age x to x + 1, called qx. In the continuous case, we could also consider the conditional probability of a person who has attained age (x) dying between ages x and x + Δx, which is
where FX(x) is the cumulative distribution function
of the continuous age-at-death random variable, X. As Δx
tends to zero, so does this probability in the continuous case. The
approximate force of mortality is this probability divided by
Δx. If we let Δx tend to zero, we get the
function for force of mortality:
Since
fX(x)=F
'X(x) is the probability density
function of X, and S(x) = 1 -
FX(x) is the survival
function, the force of mortality can also be expressed variously
as:
To understand conceptually how
the force of mortality operates within a population, consider that
the ages, x, where the probability density function
fX(x) is zero, there is no
chance of dying. Thus the force of mortality at these ages is zero.
The force of mortality μ(x) uniquely defines a
probability density function
fX(x).
The force of mortality can be interpreted as the conditional density of failure at age x, while f(x) is the unconditional density of failure at age x. The unconditional density of failure at age x is the product of the probability of survival to age x, and the conditional density of failure at age x, given survival to age x.
This is expressed in symbols as
or equivalently
In many instances, it is also desirable to determine the survival probability function when the force of mortality is known. To do this, integrate the force of mortality over the interval x to x + t
By the fundamental theorem of
calculus, this is simply
Let us denote
then taking the exponent to the base e, the survival probability of an individual of age x in terms of the force of mortality is