The Uniform Distribution of Deaths Assumption

What UDD Assumes

A life table commonly records survivors at integer ages, but a calculation may begin or end at an age such as $x+0.4$. The integer-age entries alone do not specify how deaths are distributed within the year from age $x$ to age $x+1$. The uniform distribution of deaths assumption, abbreviated UDD, supplies a particular rule for that interval: deaths are spread uniformly across it, so the number alive decreases linearly with age.

UDD is an interpolation assumption between known integer-age life-table values. It does not assert that the same mortality pattern applies across multiple years, and it does not mean that a person's entire future lifetime has a uniform distribution. The assumption is applied separately within each one-year age interval for which it is being used.

Definition: Under UDD in the age interval from $x$ to $x+1$, the number alive at fractional age $x+t$, for $0\leq t\leq 1$, decreases linearly from $\ell_x$ to $\ell_{x+1}$. Writing $d_x=\ell_x-\ell_{x+1}$ for the number of deaths during the interval, this assumption is expressed as $$ \ell_{x+t}=\ell_x-t d_x=(1-t)\ell_x+t\ell_{x+1}. $$

At $t=0$, this gives $\ell_x$; at $t=1$, it gives $\ell_{x+1}$. At $t=0.5$, it places half of the interval's deaths before age $x+0.5$, and half after. The quantities $\ell_{x+t}$ are life-table survivor counts, often based on a hypothetical cohort or radix. They need not be whole numbers at fractional ages.

From Annual Deaths to Fractional-Age Probabilities

Let $q_x$ denote the probability that a life aged $x$ dies before age $x+1$. In a life table, $q_x=d_x/\ell_x$, provided $\ell_x>0$. Under UDD, the fraction $t$ of the age interval contains the fraction $t$ of its deaths. Consequently, out of the original $\ell_x$ lives, $t d_x$ are expected to have died by age $x+t$.

{}_tq_x=\frac{t d_x}{\ell_x}=tq_x, \qquad {}_tp_x=1-{}_tq_x=1-tq_x, \qquad 0\leq t\leq 1.

Here ${}_tq_x$ is the probability that a life aged $x$ dies within the next $t$ years, and ${}_tp_x$ is the probability of surviving those $t$ years. These are probabilities measured from age $x$, not from a later fractional age. The formulas have the expected endpoints: at $t=0$, no time has passed and ${}_0q_x=0$; at $t=1$, they recover $q_x$ and $p_x=1-q_x$.

Start with the annual interval.
Find $d_x=\ell_x-\ell_{x+1}$, the deaths from age $x$ through age $x+1$, and $q_x=d_x/\ell_x$.

Take the fraction of the interval.
For age $x+t$, UDD assigns the fraction $t$ of the interval's deaths to the time from $x$ to $x+t$.

Use the intended denominator.
For a probability from age $x$, divide those deaths by $\ell_x$. For a probability conditional on being alive at a fractional age, divide by the number alive at that fractional age.

The probability from a fractional starting age requires care. For example, among lives who have reached age $x+s$, the probability of dying before age $x+t$, where $0\leq s<t\leq1$, is based on deaths in that subinterval and survivors at its start. Under the linear survivor-count assumption, that probability is $\bigl(\ell_{x+s}-\ell_{x+t}\bigr)/\ell_{x+s}$. This conditional probability is not generally $(t-s)q_x$, because its denominator is the number surviving to $x+s$, rather than the original $\ell_x$.

Why the Partial-Year Formula Holds

The justification follows directly from the meaning of uniformly distributed deaths. Conditional on dying during the year, the death time is spread uniformly over the interval of length one. Thus the probability that such a death occurs in the first $t$ years of that interval is $t$. Of the $\ell_x$ lives at age $x$, $d_x$ are expected to die during the full interval, so the number expected to die by age $x+t$ is $t d_x$. Dividing by the original $\ell_x$ gives ${}_tq_x=t d_x/\ell_x=tq_x$. Subtracting this from one gives the survival probability.

This argument explains both the proportionality and its limits. The fraction $t$ applies to deaths during the particular one-year interval, not to all deaths over a longer span. When a calculation crosses an integer age, each interval must be treated using its own life-table entries and its own UDD assumption.

Worked Applications

Worked Example: A Fraction of the Annual Deaths

A hypothetical life table has $\ell_{52}=2{,}400$ and $\ell_{53}=2{,}280$. Assume UDD from age 52 to age 53. Find the expected number of deaths by age 52.4, the number alive at age 52.4, and the probability that a life aged 52 survives to 52.4.

Method. First find the deaths in the full year: $d_{52}=2{,}400-2{,}280=120$. The fractional part of the year is $t=0.4$, so UDD assigns $0.4(120)=48$ deaths to the first 0.4 years.

The number alive at age 52.4 is $2{,}400-48=2{,}352$. Equivalently, linear interpolation gives $\ell_{52.4}=(1-0.4)(2{,}400)+0.4(2{,}280)=1{,}440+912=2{,}352$.

The annual probability is $q_{52}=120/2{,}400=0.05$. Therefore the probability of death by age 52.4, from age 52, is $0.4(0.05)=0.02$, and the survival probability is $1-0.02=0.98$.

Check. The 48 deaths are $48/2{,}400=0.02$ of the original cohort, matching the calculated death probability. The survivor and death counts add to $2{,}352+48=2{,}400$. The common mistake avoided is using the full-year $q_{52}$ as though 0.4 years were a full year.

Worked Example: Conditioning on Survival to a Fractional Age

For a hypothetical cohort, $\ell_{40}=1{,}600$ and $\ell_{41}=1{,}480$. Assume UDD during this age year. Find the probability that a life aged 40.25 dies before age 40.75, conditional on having survived to age 40.25.

Method. The annual death count is $d_{40}=1{,}600-1{,}480=120$. At age 40.25, UDD places $0.25(120)=30$ deaths before that age, leaving $1{,}600-30=1{,}570$ lives. At age 40.75, the deaths since age 40 total $0.75(120)=90$.

Thus the number of deaths between ages 40.25 and 40.75 is $90-30=60$. Conditional on being among the 1,570 lives at age 40.25, the required probability is $60/1{,}570=0.03821656\ldots$, or 0.0382 to four decimal places.

Check. The survivor count at 40.75 is $1{,}600-90=1{,}510$, and $1{,}570-1{,}510=60$, as required. A common error is to use $60/1{,}600=0.0375$, which measures deaths in this segment as a proportion of the original age-40 cohort, not as a probability conditional on reaching 40.25.

Worked Example: Recovering a Fractional-Age Survivor Count

A hypothetical table gives $\ell_{67}=900$ and $q_{67}=0.08$. Under UDD, find $\ell_{67.6}$, the probability of death between ages 67 and 67.6 for a life aged 67, and the corresponding survival probability. Give probabilities to four decimal places.

Method. The number dying during the year is $d_{67}=900(0.08)=72$. By age 67.6, UDD assigns $0.6(72)=43.2$ deaths from the original 900 lives. Therefore $\ell_{67.6}=900-43.2=856.8$.

The death probability from age 67 over 0.6 years is ${}_{0.6}q_{67}=43.2/900=0.0480$, which also equals $0.6(0.08)=0.0480$. The survival probability is ${}_{0.6}p_{67}=1-0.0480=0.9520$.

Check. The fractional-age survivor count divided by the age-67 count is $856.8/900=0.9520$, matching the survival probability. The common mistake avoided is rounding the intermediate death count to a whole life; life-table counts under an assumed interpolation need not be integers.

Interpretation, Scope, and Exam Traps

UDD is useful when a problem provides annual life-table values but asks for a quantity at a fractional age. Its central interpretation is simple: within the specified year, the proportion of annual deaths accumulated by time $t$ is $t$. In practice, first identify the exact age interval and the population to which the requested probability refers. Then distinguish an expected life-table count from a probability, and a probability from the original integer age from one conditional on survival to a fractional age.

Do not treat UDD as a statement about all future lifetime. It describes the placement of deaths within a specified one-year age interval. A separate assumption is needed for each subsequent year.
Do not confuse the two denominators. A probability from age $x$ uses $\ell_x$; a conditional probability from age $x+s$ uses $\ell_{x+s}$.
Do not confuse a count with a probability. The fractional death count is $t d_x$; the corresponding probability from age $x$ is $t d_x/\ell_x=tq_x$.
Check endpoints. At $t=0$, the survivor count must be $\ell_x$; at $t=1$, it must be $\ell_{x+1}$. The partial-year death probability must likewise start at zero and end at $q_x$.
Check the assumption. Integer-age data by themselves do not establish UDD. Use the interpolation only when the question states or otherwise specifies that assumption.

Key takeaway: Under UDD, deaths are distributed uniformly within each one-year age interval, so survivor counts decrease linearly between integer ages. From age $x$, the probability of death within the next fraction $t$ of a year is $tq_x$; conditional probabilities from a fractional age require the survivors at that fractional age as the denominator.

Check Your Understanding

For each problem, assume UDD only in the age interval identified. Keep counts unrounded during calculations and give probabilities to four decimal places.

A hypothetical table has $\ell_{30}=500$ and $\ell_{31}=460$. Find the number alive at age 30.25 and the probability, from age 30, of death by age 30.25. (Answer: $\ell_{30.25}=490$; probability $=0.0200$)
For a hypothetical cohort, $\ell_{55}=1{,}200$ and $q_{55}=0.10$. Find the expected deaths between ages 55 and 55.7 and the probability of survival from age 55 to age 55.7. (Answer: 84 deaths; survival probability $=0.9300$)
A hypothetical table has $\ell_{70}=800$ and $\ell_{71}=720$. Find the probability of death between ages 70.2 and 70.6, conditional on survival to age 70.2. (Answer: $32/784=0.0408$)
For a hypothetical cohort, $\ell_{44}=2{,}000$, $\ell_{45}=1{,}900$, and UDD applies from age 44 to 45. Find the survivor count at age 44.8 and the probability of death from age 44 by age 44.8. (Answer: $\ell_{44.8}=1{,}920$; probability $=0.0400$)
A hypothetical life table gives $\ell_{62}=1{,}500$ and $\ell_{63}=1{,}410$. Under UDD, find the probability of death between ages 62.4 and 62.9 conditional on survival to age 62.4. (Answer: $45/1{,}464=0.0307$)

What You'll Learn