Uniform Distribution of Deaths¶

Definition¶

The uniform distribution of deaths (UDD) assumption is ${}_sq_x=sq_x$ for integer age $x$ and $0\le s\le1$ $1$ . If imposed in every subsequent year, the fractional part $R_x=T_x-K_x$ is uniform on $(0,1)$ and independent of $K_x$ .

Interpretation¶

Within each year of age, deaths are spread uniformly rather than concentrated at particular dates. The assumption gives a rule for interpolating an integer-age life table.

Plot of the probability of surviving from age 60 to 60 + s under UDD with a death probability of 0.1, a straight line from 1 down to 0.9 with points marked at ages 60.5 and 60.75.

Under UDD the survival curve within a year of age is a straight line between the table values at integer ages: with $q_{60}=0.1$ $0.1$ , ${}_sp_{60}=1-0.1s$ falls from $1$ to $0.9$ . The probability of surviving from 60.5 to 60.75 is the ratio of the two marked heights, $0.925/0.95\approx0.97368$ $0.97368$ , as in the example. The vertical axis starts at $0.88$ , so the drop is magnified. The line has constant slope, but relative to the shrinking survival level that slope grows, so $\mu_{60+s}=q_{60}/(1-sq_{60})$ increases within the year.

Change $q_{60}$ , $s$ and $t$ : under UDD ${}_sp_{60}$ is a straight line, ${}_tp_{60+s}=[1-(s+t)q_{60}]/[1-sq_{60}]$ for $0\le s\le s+t\le1$ with $1-sq_{60}>0$ $0$ . The force is $q_{60}/(1-sq_{60})$ where this denominator is positive, and rises when $q_{60}>0$ $0$ . The dashed curves show constant force for comparison.

The curtate future lifetime $K_x$ identifies the year of death, and $R_x=T_x-K_x$ identifies the fraction lived within it. The uniform distribution assumption makes this fraction uniform on $(0,1)$ . As independent random variables, $R_x$ and $K_x$ permit separate averaging of fractional-year timing and the death year under UDD throughout the model.

Key Results¶

$l_{x+s}=l_x-sd_x=(1-s)l_x+sl_{x+1}$ $s l_{x + 1}$ .
$\mu_{x+s}=q_x/(1-sq_x)$ where the denominator is positive.
For $0\le s\le s+t\le1$ with $1-sq_x>0$ $0$ , ${}_tp_{x+s}=[1-(s+t)q_x]/[1-sq_x]$ $s q_{x}]$ . The positive denominator means survival to the conditioning age has positive probability.
UDD does not mean constant force within a year; its force of mortality increases when $q_x>0$ $0$ .

Examples¶

If $q_{60}=0.1$ $0.1$ , survival from age 60.5 to 60.75 is $(1-0.75\times0.1)/(1-0.5\times0.1)=0.925/0.95\approx0.97368$ $0.97368$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 2

Aliases: UDD, linear life-table interpolation, UDD assumption, uniform distribution of deaths assumption Keywords: fractional age assumption, s q x = s times q x, R x uniform and independent of K x, mu x+s = q x/(1 - s q x) Symbols: mu

Last updated: 27 October 2026 11:52