Editing Actuarial notation (section)

== Force of mortality ==

Among actuaries, '''force of mortality''' refers to what [[economists]] and other social scientists call the [[hazard rate]] and is construed as an instantaneous rate of mortality at a certain age measured on an annualized basis.

In a life table, we consider the probability of a person dying between age (''x'') and age ''x''&nbsp;+&nbsp;1; this probability is called ''q''<sub>''x''</sub>. In the continuous case, we could also consider the [[conditional probability]] that a person who has attained age (''x'') will die between age (''x'') and age (''x''&nbsp;+&nbsp;Δ''x'') as:

: <math>P_{\Delta x}(x)=P(x<X<x+\Delta\;x\mid\;X>x)=\frac{F_X(x+\Delta\;x)-F_X(x)}{(1-F_X(x))}</math>

where ''F''<sub>''X''</sub>(''x'') is the [[cumulative distribution function]] of the continuous age-at-death [[random variable]], X. As Δ''x'' tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by Δ''x''. If we let Δ''x'' tend to zero, we get the function for '''force of mortality''', denoted as ''μ''(''x''):

:<math>\mu\,(x)=\frac{F'_X(x)}{1-F_X(x)}</math>