Fractional-Year Death Timing Under UDD

Why Fractional-Year Death Timing Matters

Curtate future lifetime records the integer number of complete years lived, while complete future lifetime also records the fraction of the final year. When a problem specifies that death occurs during a particular age year, the timing within that year can determine a probability or expected age at death. Under the uniform distribution of deaths (UDD) assumption, that fraction has a particularly simple conditional distribution.

The key is to distinguish two questions: how likely someone is to die during a given year, and when within that year death occurs, given that it does occur. UDD answers the second question by assuming deaths are spread evenly through each integer age year. The annual probability of death still determines how likely the year of death is; it cancels out when we condition on death in that year.

Definition: Let $T_x$ be the complete future lifetime of a life aged $x$, and let $K_x=\lfloor T_x\rfloor$ be the curtate future lifetime. If $K_x=j$, death occurs during the age year from $x+j$ to $x+j+1$. The fractional-year death time is $U_j=T_x-j$ on that event, with $0\le U_j<1$.

The UDD Assumption Within an Age Year

Under UDD, the number of survivors decreases linearly between successive integer ages. Equivalently, for $0\le s\le1$, the probability of dying in the first fraction $s$ of the year of age $x+j$, conditional on being alive at its start, is $s q_{x+j}$. Here $q_{x+j}$ is the one-year probability of death at age $x+j$.

UDD assumption: For an integer age $x+j$ and $0\le s\le1$, $$ {}_{s}q_{x+j}=s q_{x+j}. $$ Thus, if $q_{x+j}>0$, the conditional distribution of the fractional time $U_j$, given death during that age year, is uniform on $[0,1]$.

The condition $q_{x+j}>0$ matters: if no one can die during that year according to the model, then conditioning on death during that year is not defined. If $q_{x+j}>0$, the probability of death at an exact fractional time is zero under the continuous uniform distribution, so including or excluding either endpoint does not change the probabilities below.

Why the Conditional Distribution Is Uniform

The result follows directly from conditional probability. Among lives alive at the start of age year $x+j$, the probability of dying by fraction $s$ of the year is $s q_{x+j}$. The probability of dying at some point during the full year is $q_{x+j}$. Dividing the first probability by the second gives the conditional probability of having died by fraction $s$, given death in that year.

P(U_j\le s\mid K_x=j) =\frac{{}_{s}q_{x+j}}{q_{x+j}} =\frac{s q_{x+j}}{q_{x+j}} =s,\qquad 0\le s\le1.

The cumulative distribution function $F(s)=s$ on $[0,1]$ is the cumulative distribution function of a uniform random variable on that interval. In particular, for $0\le a\le b\le1$,

P(a\le U_j\le b\mid K_x=j)=b-a.

This conditional result does not say that the probability of dying during every age year is the same. It says that, once a particular year of death is specified, the fraction of that year lived before death is uniform under UDD. The probability of reaching that year and then dying in it is still determined by the survival probability to the year and the relevant $q$.

Worked Example: Death Between Two Months of the First Year

A hypothetical life aged 40 has a one-year death probability $q_{40}=0.12$. Assume UDD in the year of age 40. Find (1) the probability of death between two and eight months after age 40, conditional on death during the year, and (2) the unconditional probability of death during that interval.

Method. Convert the month marks into fractions of a year, use the uniform conditional distribution for the interval probability, and then multiply by the probability of death during the full year to obtain the unconditional probability.

Two months is $2/12=1/6$ of a year, and eight months is $8/12=2/3$ of a year. Therefore,

P\left(\frac16\le U_0\le\frac23\mid K_{40}=0\right) =\frac23-\frac16 =\frac46-\frac16 =\frac12.

The conditional probability is $0.50$. The probability of death during the entire year is $0.12$, so the unconditional probability of death in the specified part of the year is

P\left(\frac16\le T_{40}\le\frac23\right) =0.12(0.50) =0.06.

Thus the answers are $0.50$ conditional on death in the year and $0.06$ unconditionally. As a check, the interval covers half the year, so UDD assigns it half of the annual death probability: $0.12/2=0.06$. A common mistake is to report $0.50$ as the unconditional probability; that omits the condition that the life must die during the year at all.

Using a Specified Future Year

The same reasoning applies when death is specified to occur in a later year of future lifetime. Write $K_x=j$ for death during the year beginning at age $x+j$. Conditional on reaching that year and dying within it, the fractional timing is uniform under UDD. To find an unconditional probability from age $x$, include both the probability of surviving to the start of the year and the probability of dying during that year.

P(a\le T_x-j\le b,\ K_x=j) ={}_jp_x\,q_{x+j}(b-a), \qquad 0\le a\le b\le1.

This formula is a product of three factors: survival to age $x+j$, death during the following year, and the fraction $b-a$ of that year covered by the interval. It applies under UDD in the year beginning at age $x+j$; if $q_{x+j}=0$, the unconditional probability is zero and the conditional distribution given death in that year is undefined.

Worked Example: A Fractional Interval in a Later Year

For a hypothetical life aged 50, suppose ${}_3p_{50}=0.72$ and $q_{53}=0.10$. Assume UDD during the year of age 53. Find the probability, from age 50, that death occurs between ages 53.25 and 53.80. Also find the probability of that interval conditional on survival to age 53.

Method. The specified year is the fourth year of future lifetime, so $j=3$. The fractional interval is from $0.25$ to $0.80$, with width $0.80-0.25=0.55$.

Conditional on survival to age 53, death must occur during the year of age 53 and within the specified $0.55$ fraction of that year. Under UDD, its probability is

q_{53}(0.55)=0.10(0.55)=0.055.

To obtain the probability from age 50, also require survival to age 53:

{}_3p_{50}\,q_{53}(0.55) =0.72(0.10)(0.55) =0.072(0.55) =0.0396.

The unconditional probability is $0.0396$, and the probability conditional on survival to age 53 is $0.055$. As a check, the full-year probability from age 50 of surviving to 53 and then dying during the following year is $0.72(0.10)=0.072$; the requested interval contains $55\%$ of that year, and $0.072(0.55)=0.0396$. A common mistake is to multiply by ${}_3p_{50}$ when the question asks only for a probability conditional on survival to age 53.

Conditional Moments and Quantiles

Because the fractional-year death time has a uniform distribution on $[0,1]$ conditional on death during the specified year, its standard moments and quantiles follow from that distribution. These describe timing within the year, not the probability of entering the year or dying in it.

Result: If $q_{x+j}>0$ and UDD applies in the year beginning at age $x+j$, then conditional on $K_x=j$, $U_j$ is uniform on $[0,1]$. Therefore, $$ E[U_j\mid K_x=j]=\frac12,\qquad \operatorname{Var}(U_j\mid K_x=j)=\frac1{12}. $$ For $0\le r\le1$, the conditional $r$-quantile of $U_j$ is $r$.

Since $T_x=j+U_j$ when $K_x=j$, the corresponding conditional mean and variance of complete future lifetime are

E[T_x\mid K_x=j]=j+\frac12,\qquad \operatorname{Var}(T_x\mid K_x=j)=\frac1{12}.

The conditional standard deviation is $1/\sqrt{12}$, approximately $0.289$ years. The conditional median is at fraction $0.5$, and the conditional probability of death in the last quarter of the year is $0.25$. These statements concern the distribution after conditioning on $K_x=j$; they should not be used as unconditional moments of $T_x$.

Worked Example: Conditional Age at Death and an Unconditional Interval

A hypothetical life aged 60 has ${}_4p_{60}=0.60$ and $q_{64}=0.20$. Assume UDD during the year of age 64. Given death during that year, find the mean age at death and the probability that death occurs between ages 64.60 and 64.90. Then find the unconditional probability, from age 60, of death in that interval.

Method. Given death in the year of age 64, $j=4$ and $U_4=T_{60}-4$ is uniform on $[0,1]$. Use the uniform mean for the conditional age at death, and use the interval width for the conditional interval probability.

The conditional mean fractional time is $1/2$. Therefore the mean complete future lifetime, conditional on death in that year, is

E[T_{60}\mid K_{60}=4] =4+\frac12 =4.5\text{ years}.

The age at death is $60+T_{60}$, so its conditional mean is $60+4.5=64.5$ years. The fractional interval from $0.60$ to $0.90$ has width $0.90-0.60=0.30$, giving

P(0.60\le U_4\le0.90\mid K_{60}=4)=0.30.

The unconditional probability includes survival to age 64, death during the year of age 64, and the specified part of that year:

{}_4p_{60}\,q_{64}(0.30) =0.60(0.20)(0.30) =0.12(0.30) =0.036.

Thus the conditional mean age at death is $64.5$, the conditional interval probability is $0.30$, and the unconditional interval probability from age 60 is $0.036$. As a check, the probability of reaching age 64 and dying in the next year is $0.60(0.20)=0.12$; $30\%$ of this is $0.036$. The common error is to treat the conditional mean age $64.5$ as an unconditional mean age at death, even though it is conditional on dying in a particular year.

Common Mistakes and a Reliable Procedure

Fractional-year questions are often short, but they can combine several distinct conditions. In particular, the fraction of a year in the requested interval is not automatically the probability from the original age. First identify what event is being conditioned on; then decide whether the question asks for a conditional or unconditional probability.

Identify the year of death.
Express it as $K_x=j$, so the year begins at age $x+j$.

Convert ages or months to fractions.
For ages $x+j+a$ and $x+j+b$, use fractions $a$ and $b$ within the year; the interval width is $b-a$.

Apply the conditional UDD result.
Given death during that year, the probability of the fractional interval is $b-a$.

Add earlier survival and annual death probabilities if needed.
For a probability from age $x$, multiply by ${}_jp_x q_{x+j}$.

Using interval width as an unconditional probability. A width of $0.30$ gives a conditional probability of $0.30$, given death in the year. The unconditional probability also requires the chance of reaching and dying in that year.
Using the wrong year's $q$. For $K_x=j$, the annual death probability is $q_{x+j}$, not $q_x$ unless $j=0$.
Mixing years with fractions. Ages 53.25 to 53.80 describe fractions $0.25$ to $0.80$ of the year beginning at age 53; the interval width is $0.55$, not $53.80-53.25$ interpreted as a number of years after age 50.
Conditioning on an impossible event. If $q_{x+j}=0$, there is no conditional distribution given death in that year.
Applying UDD outside its year. The linear interpolation applies within each integer age year under the UDD assumption. It does not assert that death time is uniform over several years.

Key takeaway: Under UDD, conditional on death during a specified integer age year with positive $q$, the fraction of that year lived before death is uniform on $[0,1]$. A subinterval of width $b-a$ therefore has conditional probability $b-a$; for an unconditional probability from an earlier age, multiply by the probability of surviving to and dying in the specified year.

Check Your Understanding

For each question, apply UDD only to the specified age year and distinguish conditional from unconditional probabilities.

A hypothetical life aged 35 has $q_{35}=0.06$. Given death during the year of age 35, find the probability of death between months 1 and 5. Also find the unconditional probability of death during that interval. (Answer: conditional probability $=1/3$; unconditional probability $=0.0200$)
For a hypothetical life aged 48, ${}_2p_{48}=0.81$ and $q_{50}=0.10$. Under UDD at age 50, find the probability from age 48 of death between ages 50.20 and 50.70. (Answer: $0.0405$)
Given death during the year beginning at age $x+j$, find the probability that death occurs in the last quarter of that year. (Answer: $0.25$)
Under UDD, given $K_{52}=3$ and $q_{55}>0$, find $E[T_{52}\mid K_{52}=3]$ and $\operatorname{Var}(T_{52}\mid K_{52}=3)$. (Answer: mean $=3.5$ years; variance $=1/12$ years squared)
A hypothetical life aged 70 has ${}_5p_{70}=0.50$ and $q_{75}=0.08$. Under UDD in the year of age 75, find the probability from age 70 of death between ages 75.40 and 75.90. (Answer: $0.0200$)

What You'll Learn