Start with a generic time until failure (T) that may be a function of one or more lives.
Define the curtate time until failure (K = [T]), where ([cdot]) denotes the greatest integer function. Then,
begin{eqnarray*}
Pr(K=k) = Pr( k le T < k+1) = F_T(k+1) - F_T(k).
end{eqnarray*}
Special Case. Joint Life Status. In this case (T=T(xy)), and so (K) may be denoted as (K(xy) = [T(xy)]). We may also think of the curtate random variable as (K(xy) = min(K(x), K(y))).
This has probability mass function
begin{eqnarray*}
Pr(K=k) &=& ~_{k+1} q_{xy} - ~ _k q_{xy} = ~ _k p_{xy} - ~_{k+1} p_{xy}
& =& ~ _k p_{xy} q_{x+k:y+k} equiv ~_{k|} q_{xy} .
end{eqnarray*}
For insurances, we have the first-to-die insurance
begin{eqnarray*}
A_{xy} = mathrm{E~} v^{K(xy)+1} = sum_{k=0}^{infty} v^{k+1}
~_{k|} q_{xy} .
end{eqnarray*}
This is the EPV for a payment of 1 at the end of the year of the first death among (x) and (y).
For annuities, we have
begin{eqnarray*}
ddot{a}_{xy} = mathrm{E~} ddot{a}_{overline{K(xy)+1}|} =
sum_{k=0}^{infty} v^k ~ _k p_{xy} .
end{eqnarray*}
This is the EPV for a payment of 1 at the beginning of each year while both (x) and (y) are alive.
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