Complete Lifetime Moments from UDD Year Segments

Why Conditional Death-Time Moments Matter

A life table gives the probability that a life aged $x$ dies before age $x+1$, but that annual probability does not say how deaths are distributed within the year. Under the uniform distribution of deaths assumption (UDD), that timing distribution has a particularly simple form: conditional on dying during the year, the fraction of the year lived before death is uniform on the interval from 0 to 1.

This result gives the conditional average time to death, its second raw moment, and its variance directly. The conditioning is essential. The annual probability of death, $q_x$, determines how likely it is that death occurs in the year; it does not change the distribution of timing once we know that death occurred in that year.

Definition: Let $T_x$ be the complete future lifetime of a life aged exactly $x$, measured in years. The event $T_x<1$ means death occurs during the age year from $x$ to $x+1$. Under UDD, provided $q_x>0$, the conditional death-time moments are the moments of $T_x$ given $T_x<1$.

Conditional Distribution and Moments Under UDD

Write $q_x={}_1q_x$ for the probability that a life aged $x$ dies within one year, conditional on being alive at age $x$. UDD assumes deaths are spread uniformly over that age year. Thus, for $0\leq t\leq1$, the probability of surviving from age $x$ to age $x+t$ is $1-tq_x$. Equivalently, the probability of dying by time $t$, while measured from age $x$, is $tq_x$.

Before conditioning on death in the year, the density of death time over $0<t<1$ is therefore $q_x$. The total probability of death in that interval is $\int_0^1q_x\,dt=q_x$. Dividing the density by this event probability gives the conditional density.

f_{T_x\mid T_x<1}(t) =\frac{q_x}{q_x}=1,\qquad 0<t<1,\quad q_x>0.

The conditional density is that of a uniform random variable on $(0,1)$. For any real $r>-1$, integrate $t^r$ against this density to obtain the conditional $r$-th raw moment:

E[T_x^r\mid T_x<1] =\int_0^1 t^r\,dt =\frac{1}{r+1}.

In particular, the first conditional raw moment is $1/2$ year and the second conditional raw moment is $1/3$ year squared. The conditional variance is the second raw moment minus the square of the first moment.

E[T_x\mid T_x<1]=\frac12,\qquad E[T_x^2\mid T_x<1]=\frac13,\qquad \operatorname{Var}(T_x\mid T_x<1) =\frac13-\left(\frac12\right)^2 =\frac1{12}.

These results do not depend on the value of $q_x$. If $q_x$ is small, death during the year is less likely, but among the lives that do die during that year UDD still assigns equal conditional density to every time within the year. If $q_x=0$, conditioning on death in the year is impossible, so the conditional moments are undefined.

Why the Moments Have These Values

The derivation follows from two separate steps: find the probability density of death during a small part of the year, then condition on the event that death occurs somewhere in the year. This distinction explains why $q_x$ cancels.

Use the UDD assumption.
By time $t$ in the year, the probability of death is $tq_x$. Differentiating with respect to $t$ gives the unconditional density $q_x$ on $0<t<1$.

Condition on dying in the year.
The probability of the conditioning event is $P(T_x<1)=q_x$. The conditional density is therefore $q_x/q_x=1$ throughout the year.

Integrate to obtain moments.
Integrating $t$ and $t^2$ from 0 to 1 gives $1/2$ and $1/3$. Subtracting the square of the mean from the second raw moment gives variance $1/12$.

The same result can be expressed in terms of age at death. Conditional on dying in the year, age at death is $x+T_x$, so its mean is $x+1/2$, its second raw moment is $x^2+x+1/3$, and its variance remains $1/12$. Adding a constant shifts the mean but does not change the variance.

Worked Applications

Worked Example: Moments From an Annual Death Probability

A hypothetical life table has $l_x=12{,}500$ lives at age $x$ and $l_{x+1}=9{,}500$ lives at age $x+1$. Assume UDD during the year. Given that a life aged $x$ dies before age $x+1$, calculate the conditional mean time to death, second raw moment, variance, and probability that death occurs after 0.6 years.

Method and annual probability. The number of deaths during the year is $d_x=12{,}500-9{,}500=3{,}000$. Thus $q_x=d_x/l_x=3{,}000/12{,}500=0.24$. Since $q_x>0$, conditioning on death in the year is valid.

Conditional moments. UDD gives a uniform conditional distribution on $(0,1)$, so $E[T_x\mid T_x<1]=1/2=0.5$ year and $E[T_x^2\mid T_x<1]=1/3$ year squared. The variance is $1/3-(1/2)^2=1/3-1/4=1/12\approx0.0833$ year squared.

Probability after 0.6 years. For a uniform conditional distribution, the probability of a time in an interval is its length. Therefore $P(T_x>0.6\mid T_x<1)=1-0.6=0.4$.

Check. The mean $0.5$ is the midpoint of the year, and the probability of being after 0.6 is the remaining interval length, $0.4$. The value $q_x=0.24$ affects the chance of dying in the year, not these conditional results. The common mistake is to multiply $1/2$ by $q_x$ when calculating the conditional mean.

Worked Example: Moments of Age at Death From Life-Table Values

For a hypothetical table, $l_{45}=9{,}600$ and $l_{46}=8{,}640$. Assume UDD between these ages. Conditional on dying during this age year, calculate the mean age at death, the second raw moment of age at death, and its variance.

Method and check of the conditioning event. The number of deaths is $d_{45}=9{,}600-8{,}640=960$, so $q_{45}=960/9{,}600=0.10$. Death in the year has positive probability, and the conditional time $T_{45}$ has the UDD moments $1/2$, $1/3$, and $1/12$ for its mean, second raw moment, and variance.

Mean age at death. Age at death is $45+T_{45}$. Hence $E[45+T_{45}\mid T_{45}<1]=45+1/2=45.5$ years.

Second raw moment. Expand the square before taking expectations: $E[(45+T_{45})^2\mid T_{45}<1]=45^2+2(45)(1/2)+1/3$. The terms are $2{,}025$, $45$, and $1/3$, giving $2{,}070+1/3=2{,}070.3333$ years squared.

Variance. Adding 45 years does not affect variance, so $\operatorname{Var}(45+T_{45}\mid T_{45}<1)=1/12\approx0.0833$ year squared. As a check, subtracting the squared mean from the second raw moment gives $2{,}070.3333-(45.5)^2=2{,}070.3333-2{,}070.25=0.0833$ year squared, to four decimal places.

The common mistake is to report $1/3$ as the second moment of age at death. It is the second moment of time since age 45; the age-at-death moment also includes the constant and cross-product terms.

Worked Example: A Timing-Based Amount at Death

A hypothetical contract pays $Z=1{,}200(1-T_x)$ dollars if death occurs during the year of age $x$, where $T_x$ is measured in years from age $x$. Assume UDD. Conditional on death during the year, calculate the expected payment and the payment variance.

Method. First find the moments of the time-dependent payment using the conditional moments of $T_x$. The payment decreases linearly from $1{,}200$ dollars at the start of the year to 0 dollars at its end.

Expected payment. $E[Z\mid T_x<1]=1{,}200(1-E[T_x\mid T_x<1])=1{,}200(1-1/2)=600$ dollars.

Second raw moment. Square the payment expression: $Z^2=1{,}440{,}000(1-2T_x+T_x^2)$. Therefore $E[Z^2\mid T_x<1]=1{,}440{,}000(1-2(1/2)+1/3)=1{,}440{,}000/3=480{,}000$ dollars squared.

Variance and check. The variance is $480{,}000-600^2=480{,}000-360{,}000=120{,}000$ dollars squared. As a separate check, $Z=1{,}200-1{,}200T_x$, so its variance is $1{,}200^2(1/12)=120{,}000$ dollars squared. Both calculations agree.

The common mistake is to treat the second raw moment as the variance. Here $480{,}000$ is the second raw moment; subtracting the squared mean is necessary to obtain the variance.

Common Conditioning and Moment Errors

The main exam trap is to confuse the probability of the conditioning event with a moment conditional on that event. Under UDD, $q_x$ is the probability of dying during the age year. Once death during that year is given, the conditional density is 1, not $q_x$.

Multiplying by $q_x$ after conditioning. The expression $q_x/2$ is not $E[T_x\mid T_x<1]$. It combines an event probability with a conditional mean. The conditional mean is $1/2$ year.
Confusing raw moment and variance. The conditional second raw moment is $1/3$; the conditional variance is $1/3-(1/2)^2=1/12$.
Mixing up time and age. $T_x$ is elapsed time since age $x$, whereas age at death is $x+T_x$. Their means and second raw moments differ, though their variances are equal.
Ignoring the event specified. These moments condition on death during this particular age year. They are not moments of complete future lifetime without that condition.

Key takeaway: Under UDD, conditional on death in the age year, time to death is uniform on $(0,1)$. Its first raw moment is $1/2$, its second raw moment is $1/3$, and its variance is $1/12$, regardless of $q_x>0$.

Check Your Understanding

For each problem, condition on the event stated and use UDD only within the specified integer age year.

A life aged 50 has $q_{50}=0.16$. Given death before age 51, find the conditional mean of $T_{50}$, its second raw moment, and its variance. (Answer: $0.5$ year; $1/3$ year squared; $1/12$ year squared.)
A hypothetical table has $l_{30}=10{,}000$ and $l_{31}=8{,}500$. Given death during the year, find the mean age at death and its variance. (Answer: $30.5$ years; $1/12$ year squared.)
Given death in the year under UDD, find $P(0.2<T_x<0.7\mid T_x<1)$ and the conditional mean of $T_x$ given $0.2<T_x<0.7$. (Answer: $0.5000$; $0.45$ year.)
A hypothetical payment is $Z=900(1-T_x)$ dollars if death occurs during the year. Given death in that year, find $E[Z]$ and $\operatorname{Var}(Z)$. (Answer: $\$450$; $\$67{,}500$ dollars squared.)
Of 20,000 lives aged $x$, suppose $q_x=0.30$, deaths during the year are subject to UDD, and each death time is measured from age $x$. Find the expected number of deaths and the expected sum of their death times in years. (Answer: 6,000 deaths; 3,000 years.)

What You'll Learn