The UDD Relationship Between Curtate and Complete Means

Why the Two Future-Lifetime Means Differ

The curtate future lifetime records only the number of complete years lived after age $x$. The complete future lifetime also includes the fraction of the final year lived. Under the uniform distribution of deaths (UDD) assumption, that final-year fraction has mean one-half year, conditional on the year of death. This gives a simple relationship between the two means—but only after the weights for the possible death years are handled correctly.

This tutorial derives the relationship by conditioning on the integer year of death, then uses it with annual mortality rates and life-table counts. The result is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: For a life aged $x$, let $T_x$ be the complete future lifetime, measured in years from age $x$. The curtate future lifetime is $K_x=\lfloor T_x\rfloor$, the greatest integer not exceeding $T_x$. Thus $K_x=k$ means the life dies between durations $k$ and $k+1$, and $T_x-K_x$ is the fraction of that final year lived.

Write ${}_k p_x$ for the probability of surviving $k$ years from age $x$, and $q_{x+k}$ for the probability of dying within the following year, conditional on being alive at age $x+k$. The unconditional probability of dying in the $k$th year after age $x$ is therefore

P(K_x=k)={}_k p_x q_{x+k}, \qquad k=0,1,2,\ldots

Here ${}_0p_x=1$, so the probability of dying during the first year is $q_x$. For the relationship derived below, assume death eventually occurs with probability one and the means are finite. The same expectation argument also applies when the nonnegative means are infinite.

What UDD Implies Within a Year of Death

Under UDD, the number of deaths in a year is spread uniformly across that year. In survival-probability terms, for $0\leq s\leq1$, the probability of surviving $k+s$ years from age $x$ is

{}_{k+s}p_x={}_k p_x(1-sq_{x+k}).

If $q_{x+k}>0$, the probability of dying during the first fraction $s$ of the year, given survival to its start, is $s q_{x+k}$. Given both survival to that start and death by the end of the year, the conditional probability of dying in that first fraction is consequently

\frac{s q_{x+k}}{q_{x+k}}=s.

That is the distribution function of a uniform random variable on $[0,1]$. Therefore, conditional on dying in year $k$, the fraction $T_x-k$ of that year lived has mean $1/2$. If $q_{x+k}=0$, there is no probability of dying in that year, so it contributes no term to the conditioning calculation.

Theorem: If UDD applies within each year, death eventually occurs with probability one, and the means are finite, then $$ E[T_x]=E[K_x]+\frac{1}{2}. $$ The additional one-half year is the average fraction of the year of death lived, averaged over all possible years of death.

Derivation by Conditioning on the Year of Death

Let $p_k=P(K_x=k)={}_k p_xq_{x+k}$. Since the probabilities $p_k$ sum to one, the law of total expectation gives

\begin{aligned} E[T_x] &=\sum_{k=0}^{\infty}P(K_x=k)E[T_x\mid K_x=k]\\ &=\sum_{k=0}^{\infty}p_k\left(k+E[T_x-k\mid K_x=k]\right). \end{aligned}

Under UDD, the conditional mean of the fraction of the year lived is $1/2$, so $E[T_x-k\mid K_x=k]=1/2$ whenever $p_k>0$. Substituting this into the sum gives

\begin{aligned} E[T_x] &=\sum_{k=0}^{\infty}p_k\left(k+\frac12\right)\\ &=\sum_{k=0}^{\infty}kp_k+\frac12\sum_{k=0}^{\infty}p_k\\ &=E[K_x]+\frac12. \end{aligned}

The last equality uses $\sum_{k=0}^{\infty}p_k=1$. This also explains why the adjustment is one-half year, not one-half year multiplied by the probability of surviving one or more years: every life dies in exactly one of the possible years, and the conditional fractional-year mean is one-half in whichever year that is.

Worked Calculations

Worked Example: Annual Mortality Rates with a Certain Final Year

A hypothetical life aged $x$ has annual death probabilities $q_x=0.20$, $q_{x+1}=0.25$, $q_{x+2}=0.40$, and $q_{x+3}=1$. Assume UDD within each year. Calculate $E[K_x]$ and $E[T_x]$. Identify a common error in weighting the death years.

Method. First calculate the unconditional probability of death in each year. Each annual $q$ is conditional on reaching the start of that year, so multiply it by the probability of surviving to that start.

The probability of death in year 0 is

$$ P(K_x=0)=1(0.20)=0.20. $$

The probability of surviving one year is $1-0.20=0.80$. Thus the probability of death in year 1 is

$$ P(K_x=1)=0.80(0.25)=0.20. $$

The probability of surviving two years is $0.80(1-0.25)=0.80(0.75)=0.60$. Hence

$$ P(K_x=2)=0.60(0.40)=0.24. $$

The probability of surviving three years is $0.60(1-0.40)=0.60(0.60)=0.36$, and everyone then dies during year 3:

$$ P(K_x=3)=0.36(1)=0.36. $$

Check the probabilities: $0.20+0.20+0.24+0.36=1.00$. The curtate mean is

\begin{aligned} E[K_x] &=0(0.20)+1(0.20)+2(0.24)+3(0.36)\\ &=0+0.20+0.48+1.08\\ &=1.76\text{ years}. \end{aligned}

By the UDD relationship,

E[T_x]=1.76+0.50=2.26\text{ years}.

Check. Weighting each year’s midpoint by its unconditional death probability gives $0.20(0.5)+0.20(1.5)+0.24(2.5)+0.36(3.5)=0.10+0.30+0.60+1.26=2.26$ years, agreeing with the relationship.

Common mistake avoided. The annual rates $0.20,0.25,0.40,1$ are conditional death probabilities, not the probabilities of each death year. Using them directly as weights would give values that do not form a probability distribution. The correct weights are $0.20,0.20,0.24,0.36$, which incorporate survival to each year.

Worked Example: Life-Table Counts and the Initial-Cohort Denominator

A hypothetical life table has $l_x=1{,}000$, $l_{x+1}=900$, $l_{x+2}=720$, $l_{x+3}=360$, and $l_{x+4}=0$. Assume UDD within each year. Find the two future-lifetime means from age $x$.

Method. The number dying in each year is the difference between consecutive life-table counts. To obtain the unconditional probability of death in a given year for a life selected at age $x$, divide that year’s deaths by the initial count $l_x=1{,}000$.

The death counts are $1{,}000-900=100$, $900-720=180$, $720-360=360$, and $360-0=360$. Dividing each by $1{,}000$, the unconditional probabilities of death in years $0,1,2,3$ are $0.10,0.18,0.36,0.36$. Their sum is $0.10+0.18+0.36+0.36=1.00$, as required.

The curtate mean is

\begin{aligned} E[K_x] &=0(0.10)+1(0.18)+2(0.36)+3(0.36)\\ &=0+0.18+0.72+1.08\\ &=1.98\text{ years}. \end{aligned}

Applying UDD gives $E[T_x]=1.98+0.50=2.48$ years. As a check, the year-of-death midpoints give $0.10(0.5)+0.18(1.5)+0.36(2.5)+0.36(3.5)=0.05+0.27+0.90+1.26=2.48$ years.

Common mistake avoided. Dividing each year’s death count by the number alive at the start of that year gives the conditional rates $0.10,0.20,0.50,1$, not the unconditional death-year probabilities. Those rates do not sum to one and cannot be used directly as the weights in $E[K_x]$.

Worked Example: Constant Annual Mortality over an Unbounded Horizon

A hypothetical life has the same annual death probability $q=0.25$ at every future integer age. Assume UDD within every year. Calculate $E[K_x]$ and $E[T_x]$, showing how the infinite series is evaluated.

Method. Let $r=1-q=0.75$ be the probability of surviving one year. The probability of surviving $k$ years is $r^k$, so $P(K_x=k)=r^kq$ for $k=0,1,2,\ldots$. These probabilities sum to $q/(1-r)=0.25/0.25=1$, confirming eventual death with probability one.

For a nonnegative integer-valued random variable, its mean is the sum of its positive-integer tail probabilities. Here $P(K_x\geq k)=r^k$ for $k\geq1$. Therefore

\begin{aligned} E[K_x] &=\sum_{k=1}^{\infty}P(K_x\geq k)\\ &=\sum_{k=1}^{\infty}(0.75)^k\\ &=\frac{0.75}{1-0.75}\\ &=3\text{ years}. \end{aligned}

The UDD relationship then gives $E[T_x]=3+0.5=3.5$ years. As a check using the death-year probabilities, $E[K_x]=q\sum_{k=0}^{\infty}kr^k=q r/(1-r)^2=0.25(0.75)/(0.25)^2=3$, which agrees.

Common mistake avoided. The year-0 death probability is $q$, while the year-$k$ probability is $r^kq$. Treating each year as having unconditional death probability $q$ would assign total probability greater than one over an unlimited horizon.

Interpretation and Exam Traps

The relationship is useful because it allows one mean to be obtained from the other without rebuilding the full distribution of death within each year. It is specific to UDD: the conditional fractional part of the year lived has mean $1/2$ under that assumption. Do not automatically apply the same adjustment under another fractional-age assumption.

The adjustment is measured in years, matching the time unit used for both $K_x$ and $T_x$. If the problem asks for months, the one-half-year adjustment is six months. Keep the distinction between conditional annual mortality and unconditional death-year probabilities explicit, especially when working from $q_x$ values or life-table counts.

Do not add one-half year for each year survived. The extra fraction belongs to the single year of death.
Do not use annual $q$-values directly as death-year weights. Multiply by the probability of surviving to the start of each year.
Do not divide all life-table death counts by different denominators when constructing the unconditional distribution from the original age $x$. Divide by the initial cohort count.
Check the probability total. The unconditional probabilities of all possible death years should sum to one when death eventually occurs with probability one.

Key takeaway: Under UDD, conditional on any particular year of death, the average fraction of that year lived is $1/2$. Averaging over the mutually exclusive death years therefore gives $E[T_x]=E[K_x]+1/2$, provided the death-year probabilities are correctly weighted.

Check Your Understanding

For each problem, assume UDD within every year and use years as the time unit. Calculate the unconditional death-year probabilities before finding the means.

A hypothetical life has $q_x=0.30$, followed by $q_{x+1}=1$. Find the probabilities of death in years 0 and 1, then find $E[K_x]$ and $E[T_x]$. (Answer: $P(K_x=0)=0.30$, $P(K_x=1)=0.70$; $E[K_x]=0.70$ years; $E[T_x]=1.20$ years.)
For a hypothetical life, the annual death probabilities are $0.40$, $0.50$, and $1$, in that order. Find the three unconditional death-year probabilities and both means. (Answer: $0.40, 0.30, 0.30$; $E[K_x]=0.90$ years; $E[T_x]=1.40$ years.)
A life table for a hypothetical group has $l_x=800$, $l_{x+1}=640$, $l_{x+2}=320$, and $l_{x+3}=0$. Use the initial count to find the death-year probabilities and the two means. (Answer: $0.20, 0.40, 0.40$; $E[K_x]=1.20$ years; $E[T_x]=1.70$ years.)
From age $x+1$, a hypothetical life has annual death probabilities $0.20$, $0.50$, and $1$, in that order. Measured from age $x+1$, find the unconditional probabilities of death in each year and the two means. (Answer: $0.20, 0.40, 0.40$; $E[K_{x+1}]=1.20$ years; $E[T_{x+1}]=1.70$ years.)
A hypothetical life has constant annual death probability $q=0.20$ at every future integer age. Find the curtate mean using the geometric survival tail, then apply UDD to obtain the complete mean. (Answer: $E[K_x]=4.00$ years; $E[T_x]=4.50$ years.)

What You'll Learn