Complete and Curtate Expectation of Life¶

Definition¶

The complete expectation of life at age $x$ is $\mathring e_x=\mathbb{E}[T_x]$ $E [T_{x}]$ , whereas the curtate expectation is $e_x=\mathbb{E}[K_x]$ $E [K_{x}]$ . Their $n$ -year term versions are $\mathbb{E}[\min(T_x,n)]$ and $\mathbb{E}[\min(K_x,n)]$ , respectively.

Interpretation¶

The complete quantity includes fractions of the final year. The curtate quantity counts only completed years. A term expectation stops accumulating lifetime after a fixed horizon.

Survival curve exp(-0.1t) shown to time 30; the full area to infinity is 10, all unit-width bars sum to about 9.508, and all gaps total about 0.492.

The figure uses the Examples force $\lambda=0.1$ $0.1$ , so ${}_tp_x=\mathrm{e}^{-0.1t}$ (drawn up to $t=30$ $30$ ). The full area to infinity is $\mathring e_x=10$ $10$ ; the visible part is $10(1-\mathrm{e}^{-3})\approx9.5021$ $9.5021$ . The bars have width one and take the height ${}_kp_x$ at the right end of each year, so the sum over all years is $\displaystyle e_x=\sum_{k\ge1}{}_kp_x\approx9.508$ $9.508$ . Only the first 30 bars are shown, with total $\displaystyle \sum_{k=1}^{30}\mathrm{e}^{-0.1k}\approx9.0349$ $9.0349$ . The red gaps between the bars and the curve are the fractional final years that the curtate count drops; over all years they total about $0.492$ (about $0.4672$ in the displayed interval), within the bound $0\le\mathring e_x-e_x\le1$ $1$ .

Change $\lambda$ : $\mathring e_x$ is the area under ${}_tp_x$ , $e_x$ the total area of the unit bars, and the red gaps add up to less than 1. Switch on the term version to stop both at $n$ years. The displayed window is finite; compare its shown contribution with the full lifetime value and omitted tail in the readout.

The curtate future lifetime $K_x$ counts complete future years. Summing its tail probabilities therefore gives the curtate expectation, whereas integrating the tails of $T_x$ includes the final fractional year.

Key Results¶

$\mathring e_x=\int_0^{\infty}{}_tp_x\,\mathrm{d}t$ and $\mathbb{E}[\min(T_x,n)]=\int_0^n{}_tp_x\,\mathrm{d}t$ $\int_{0}^{n}_{t} p_{x} d t$ .
$\displaystyle e_x=\sum_{k=1}^{\infty}{}_kp_x$ and $\displaystyle \mathbb{E}[\min(K_x,n)]=\sum_{k=1}^n{}_kp_x$ for integer $n$ .
When expectations are finite, $0\le\mathring e_x-e_x\le1$ $1$ . Under uniform distribution of deaths (UDD) at integer age the difference is $1/2$ .
$\mathbb{E}[T_x^2]=2\int_0^\infty t\,{}_tp_x\,\mathrm{d}t$ when finite.

Examples¶

At force $\lambda=0.1$ $0.1$ , the complete expectation is 10 years, while the curtate expectation is $1/(\mathrm{e}^{0.1}-1)\approx9.508$ years.

Prerequisites¶

Sources¶

Life Insurance · Lecture 1

Aliases: life expectancy, term expectation of life, complete expectation of life, curtate expectation of life, expected future lifetime, mean future lifetime, expected remaining lifetime, mean residual life Keywords: e circle x, integral of t p x, sum of k p x, difference at most one year, one half under UDD Symbols: infinity, integral, lambda, sum

Last updated: 27 October 2026 11:52