Life Tables

In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of death"). In other words, it represents the survivorship of people from a certain population. They can also be explained as a long-term mathematical way to measure a population's longevity. Tables have been created by demographers including John Graunt, Reed and Merrell, Keyfitz, and Greville.

There are two types of life tables used in actuarial science. The period life table represents mortality rates during a specific time period for a certain population. A cohort life table, often referred to as a generation life table, is used to represent the overall mortality rates of a certain population's entire lifetime. They must have had to be born during the same specific time interval. A cohort life table is more frequently used because it is able to make a prediction of any expected changes in the mortality rates of a population in the future. This type of table also analyzes patterns in mortality rates that can be observed over time. Both of these types of life tables are created based on an actual population from the present, as well as an educated prediction of the experience of a population in the near future. In order to find the true life expectancy average, 100 years would need to pass and by then finding that data would be of no use as healthcare is continually advancing.

Other life tables in historical demography may be based on historical records, although these often undercount infants and understate infant mortality, on comparison with other regions with better records, and on mathematical adjustments for varying mortality levels and life expectancies at birth.

From this starting point, a number of inferences can be derived.

• The probability of surviving any particular year of age
• The remaining life expectancy for people at different ages

Background

There are two types of life tables:

Type	Description
Period or static life tables	Show the current probability of death (for people of different ages, in the current year)
Cohort life tables	Show the probability of death of people from a given cohort (especially birth year) over the course of their lifetime

U.S. Social Security Administration (SSA) "Actuarial life table" allows study of life expectancy as a function of age already achieved.

SSA life table data, plotted to show remaining life expectancy—the number of years of life expected beyond subject's current age

Static life tables sample individuals assuming a stationary population with overlapping generations. "Static life tables" and "cohort life tables" will be identical if population is in equilibrium and environment does not change. If a population were to have a constant number of people each year, it would mean that the probabilities of death from the life table were completely accurate. Also, an exact number of 100,000 people were born each year with no immigration or emigration involved. "Life table" primarily refers to period life tables, as cohort life tables can only be constructed using data up to the current point, and distant projections for future mortality.

Life tables can be constructed using projections of future mortality rates, but more often they are a snapshot of age-specific mortality rates in the recent past, and do not necessarily purport to be projections. For these reasons, the older ages represented in a life table may have a greater chance of not being representative of what lives at these ages may experience in future, as it is predicated on current advances in medicine, public health, and safety standards that did not exist in the early years of this cohort. A life table is created by mortality rates and census figures from a certain population, ideally under a closed demographic system. This means that immigration and emigration do not exist when analyzing a cohort. A closed demographic system assumes that migration flows are random and not significant, and that immigrants from other populations have the same risk of death as an individual from the new population. Another benefit from mortality tables is that they can be used to make predictions on demographics or different populations.

However, there are also weaknesses of the information displayed on life tables. One being that they do not state the overall health of the population. There is more than one disease present in the world, and a person can have more than one disease at different stages simultaneously, introducing the term comorbidity. Therefore, life tables also do not show the direct correlation of mortality and morbidity.

The life table observes the mortality experience of a single generation, consisting of 100,000 births, at every age number they can live through.

Life tables are usually constructed separately for men and for women because of their substantially different mortality rates. Other characteristics can also be used to distinguish different risks, such as smoking status, occupation, and socioeconomic class.

Life tables can be extended to include other information in addition to mortality, for instance health information to calculate health expectancy. Health expectancies such as disability-adjusted life year and Healthy Life Years are the remaining number of years a person can expect to live in a specific health state, such as free of disability. Two types of life tables are used to divide the life expectancy into life spent in various states:

Type	Description
Multi-state life tables (also known as increment-decrements life tables)	Based on transition rates in and out of the different states and to death
Prevalence-based life tables (also known as the Sullivan method)	Based on external information on the proportion in each state. Life tables can also be extended to show life expectancies in different labor force states or marital status states.

Life tables that relate to maternal deaths and infant moralities are important, as they help form family planning programs that work with particular populations. They also help compare a country's average life expectancy with other countries. Comparing life expectancy globally helps countries understand why one country's life expectancy is rising substantially by looking at each other's healthcare, and adopting ideas to their own systems.

The mathematics

In a life table, the values of [math]q_x[/math] and other functions for different (integer) ages [math]x[/math] are tabulated. The values are assumed to be based on the survival distribution discussed in previous sections. In this section, we will discuss more functions appearing in a life table.

Suppose there are [math]\ell_0[/math] newborns. Let the indicator function

[[math]]\mathbf 1_j(x)=\begin{cases}1,&\text{if life }j\text{ survives to age }x;\\ 0,&\text{otherwise}. \end{cases}[[/math]]

Also let [math]\mathcal L(x)[/math] be the sum of all such indicator functions [math]\mathbf 1_j(x)[/math], i.e. [math]\mathcal L(x)=\sum_{j=1}^{\ell_0}\mathbf 1_j(x)[/math]. We can interpret [math]\mathcal L(x)[/math] as the number of survivors to age [math]x[/math] for the [math]\ell_0[/math] newborns. We denote the expected value of [math]\mathcal L(x)[/math] by [math]\ell_x[/math].

Proposition

[[math]]\ell_x=\ell_0S_0(x).[[/math]]

Show Proof

Since

[[math]]\mathbb E[\mathbf 1_j(x)]=1\cdot\mathbb P(\text{life }j\text{ survives to age }x)+0\cdot\mathbb P(\text{life }j\text{ does not survive to age }x)=\mathbb P(\text{life }j\text{ survives to age }x)=S_0(x)[[/math]]

(this is true for each life [math]j[/math], since the survival distribution for the future lifetime of different lifes are assumed to be the same), [math]\ell_x[/math] equals

[[math]]\mathbb E[\mathcal L(x)]=\mathbb E\left[\sum_{j=1}^{\ell_0}\mathbf 1_j(x)\right]=\sum_{j=1}^{\ell_0}\mathbb E[\mathbf 1_j(x)]=\sum_{j=1}^{\ell_0}\underbrace{S_0(x)}_{\text{constant with respect to }j}=\ell_0S_0(x)[[/math]]

.

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As a corollary,

[[math]]\ell'_x=(\ell_0S_0(x))'=\ell_0S'_0(x)[[/math]]

([math]\ell_0[/math] is constant with respect to [math]x[/math]). Also,

[[math]]\frac{-\ell'_x}{\ell_x}=\frac{-\ell_0S'_0(x)}{\ell_0S_0(x)}=\frac{-S'_0(x)}{S_0(x)}=\mu_x[[/math]]

. Also, we can use [math]\ell_x[/math] to calculate probabilities like [math]_t p_x[/math] and [math]_t q_x[/math], as follows:

[[math]]_t p_x=\frac{S_0(x+t)}{S_0(x)}=\frac{\ell_0S_0(x+t)}{\ell_0S_0(x)}=\frac{\ell_{x+t}}{\ell_x}[[/math]]

, and thus

[[math]]_t q_x=1-{}_t p_x=1-\frac{\ell_{x+t}}{\ell_x}[[/math]]

. In a later section in which selection age is involved in the life table (select table), we will use these formulas to calculate these probabilities from such life table, to incorporate the effect of selection.

We have discussed about the number of survivors to age [math]x[/math], and we will discuss the "opposite thing" in the following, namely the number of deaths to age [math]x[/math] (i.e. between age 0 and [math]x[/math]), or in general, between age [math]x[/math] and [math]x+n[/math].

We denote the expected value of such number of deaths by [math]_n d_x[/math].

Proposition

[[math]]_n d_x=\ell_x-\ell_{x+n}.[[/math]]

Show Proof

We can define another indicator function for this context similarly (with value 1 if life [math]j[/math] dies between age [math]x[/math] and age [math]x+n[/math] and 0 otherwise). Then, the expected value of each indicator function equals the probability for one of the newborns to die between [math]x[/math] and [math]x+n[/math], which is [math]S_0(x)-S_0(x+n)[/math]. With similar reasoning as in above, we have

[[math]]_n d_x=\ell_0(S_0(x)-S_0(x+n))=\ell_0S_0(x)-\ell_0S_0(x+n)=\ell_x-\ell_{x+n}.[[/math]]

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Similarly, we write [math]_1 d_x[/math] as [math]d_x[/math] for simplicity.

Apart from the life table functions [math]\ell_x[/math] and [math]_n d_x[/math] which are related to the expectation of number of survivors and deaths respectively, we will also discuss two more life table functions, that is related to the expectation of lifetime.

Example

Suppose [math]\ell_0=1000[/math] and the survival function for newborn is [math]S_0(x)=1-\frac{x}{80},\quad 0\le x\le 80[/math].

(a) What is [math]\ell_{80}[/math]?

(b) What are [math]_{50}d_{30}[/math] and [math]_{50}d_{20}[/math]? Are they equal?

Solution

(a) [math]\ell_{80}=\ell_0S_0(80)=1000(0)=0[/math]. (That is, the expected number of survivors to age 80 is 0.)

(b) [math]_{50}d_{30}=\ell_{30}-\ell_{80}=1000(1-30/80)-0=625[/math] and [math]_{50}d_{20}=\ell_{20}-\ell_{70}=1000(1-20/80)-1000(1-70/80)=625[/math]. They are equal.

There are two types of expectation of life: one is discrete and another is continuous, and they are called curtate-expectation-of-life and complete-expectation-of-life respectively.

Definition (Complete-expectation-of-life)

The complete-expectation-of-life of [math](x)[/math], denoted by [math]\overset{\circ}{e}_x[/math], is [math]\mathbb E[T_x][/math].

Proposition

[[math]]\overset{\circ}e_x=\int_{0}^{\infty}{}_t p_x\,dt.[[/math]]

Show Proof

We will use integration by parts.

[[math]]\overset{\circ}e_x=\mathbb E[T_x]=\int_{0}^{\infty}tf_x(t)\,dt =\int_{0}^{\infty}t\,d(F_x(t)) =\int_{0}^{\infty}t\,d(_t q_x) =\int_{0}^{\infty}t\,d(1-{}_t p_x) =\int_{0}^{\infty}t\,d(-{}_t p_x) =[t(-{}_t p_x)]^{\infty}_0-\int_{0}^{\infty}-{}_t p_x\,dt =\lim_{t\to \infty}(-t \;_t p_x)+\int_{0}^{\infty}{}_t p_x\,dt [[/math]]

Now, it suffices to prove that [math]\lim_{t\to \infty}(-t \;_t p_x)=0[/math] and this is true since [math]-t\to-\infty[/math] and [math]_tp_x=S_x(t)\to 0[/math] as [math]t\to\infty[/math], so this limit either equals [math]-\infty[/math] or 0. However, since the expected value exists (i.e. does not tend to infinity, or else the expectation of life does not make sense), this limit cannot equal [math]-\infty[/math], and so this limit is 0.

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• With a similar proof, we can also prove that [math]\mathbb E[T_x^2]\overset{\text{ def }}=\int_{0}^{\infty}t^2 f_x(t)\,dt=\int_{0}^{\infty}{}_t p_x\,d({t^2})=2\int_{0}^{\infty}{t}\;_t p_x\,dt[/math].
• Thus, the variance [math]\operatorname{Var}(T_x)=2\int_{0}^{\infty}t\;_t p_x\,dt-(\overset{\circ}{e}_x)^2[/math].

Proposition

[[math]]e_x=\sum_{k=1}^{\infty}{}_k p_x.[[/math]]

Show Proof

The previous proposition about [math]\overset{\circ}{e}_x[/math] uses integration by parts in the proof, and we can analogously use summation by parts (may be interpreted as a discrete analogue of integration by parts) in the proof. However, there is a simpler way to prove this proposition, where the summation is "split" appropriately:

[[math]] \begin{align} e_x&=\mathbb E[K_x]\\ &=\sum_{k=0}^{\infty}k{}_k p_x q_{x+k}\\ &=\sum_{k=0}^{\infty}k({}_k p_x-{}_{k+1}p_x)\\ &=\sum_{k={\color{darkgreen}1}}^{\infty}k{}_k p_x-\sum_{k=0}^{\infty}k{}_{k+1}p_x&(k{}_kp_x=0\text{ when }k=0)\\ &=\sum_{k={\color{darkgreen}1}}^{\infty}k{}_k p_x-\sum_{k'=1}^{\infty}(k'-1){}_{k'}p_x&(k'=k+1)\\ &=\underbrace{\sum_{k={\color{darkgreen}1}}^{\infty}k{}_k p_x-\sum_{k'=1}^{\infty}k' {}_{k'}p_x}_{=0}+\sum_{k'=1}^{\infty}{}_{k'}p_x&(\text{the first two sums represent the same thing})\\ &=\sum_{k'=1}^{\infty}{}_{k'}p_x. \end{align} [[/math]]

We can observe that this sum and the sum in the proposition represent the same thing, and thus the result follows.

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• With a similar proof, we can also prove that
  [[math]]\mathbb E[K_x^2]=\sum_{k=0}^{\infty}k^2\Delta(-{}_k p_x)=\underbrace{[k^2(-{}_k p_x)]_0^{\infty}}_{=0}-\sum_{k=0}^{\infty}{}_{k+1}p_x\underbrace{\Delta ({\color{darkgreen}k^2})}_{=k^2+2k+1-k^2=2k+1}=\sum_{k=0}^{\infty}({\color{darkgreen}2k+1}){}_{k+1}p_x=\sum_{{\color{darkgreen}k=1}}^{\infty}({\color{darkgreen}2k-1}){}_kp_x[[/math]]
  (for summation by parts method).
• Thus, the variance [math]\operatorname{Var}(K_x)=\left(\sum_{k=1}^{\infty}(2k-1){}_kp_x\right)-(e_x)^2[/math].

The following are recursion relations for [math]\overset{\circ}{e}_x[/math] and [math]e_x[/math], which can be useful when we want to find the complete/curtate-expectation-of-life of [math](x)[/math] given the expectation of a life with some other ages, say [math]x+1[/math] and [math]x+2[/math]. We will state the recursion relations as a form of proposition, and then prove them formally. After the proof, we will try to give some intuitive explanations about the recursion relation for [math]e_x[/math].

Proposition

[[math]]e_x=p_x(1+e_{x+1}).[[/math]]

Show Proof

[[math]]e_x=\sum_{k=1}^{\infty}{}_kp_x=p_x+\sum_{k=2}^{\infty}{}_kp_x=p_x+\sum_{k=2}^{\infty}p_x{}_{k-1}p_{x+1}=p_x+p_x\sum_{k=1}^{\infty}{}_kp_{x+1}=p_x(1+e_{x+1}).[[/math]]

In particular, we have

[[math]]_kp_x=\frac{S_0(x+k)}{S_0(x)}=\frac{S_0(x+1)}{S_0(x)}\cdot\frac{S_0(x+k)}{S_0(x+1)}=p_x\;_{k-1}p_{x+1}[[/math]]

.

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An intuitive explanation of this recursion relation is as follows:

• for LHS, [math]e_x[/math] is the curtate-expectation-of-life of [math](x)[/math];
• for RHS, [math]e_{x+1}[/math] is the curtate-expectation-of-life of [math](x+1)[/math], and we want to "transform" it to the expectation of [math](x)[/math]. The first step is adding 1 to it, since this is the expectation with respect to [math](x+1)[/math], but we want the expectation from the perspective of [math](x)[/math], which is 1 year younger. But only this step is not enough, since "[math]e_{x+1}[/math]" assumes the life already lives for [math]x+1[/math] years, but for [math]e_x[/math], the life is only assumed to live for [math]x[/math] years. Hence, we also need to multiply the probability for [math](x)[/math] to live for one year, [math]p_x[/math], to "get to" [math]e_{x+1}[/math].

• Now, "the expectation of life from age [math]x+1[/math] onward" is done through [math]e_{x+1}[/math]. How about "the expectation of life from age [math]x[/math] to age [math]x+1[/math]"? Indeed, when the life dies within age [math]x[/math] and [math]x+1[/math], [math]K=0[/math]. This means such "expectation of life" is zero.

Proposition

[[math]]\overset{\circ}e_x=\int_{0}^{1}{}_tp_x\,dt+p_x\overset{\circ}e_{x+1}.[[/math]]

Show Proof

[[math]] \begin{align} \overset{\circ}e_x&=\int_{0}^{\infty}{}_tp_x\,dt\\ &=\int_{0}^{1}{}_tp_x\,dt+\int_{1}^{\infty}{}_tp_x\,dt\\ &=\int_{0}^{1}{}_tp_x\,dt+p_x\int_{1}^{\infty}{}_{t-1}p_{x+1}\,dt\\ &=\int_{0}^{1}{}_tp_x\,dt+p_x\int_{{\color{darkgreen}0}}^{\infty}{}_{{\color{darkgreen}u}}p_{x+1}\,d{\color{darkgreen}u}&(u=t-1)\\ &=\int_{0}^{1}{}_tp_x\,dt+p_x\overset{\circ}e_{x+1}.\\ \end{align} [[/math]]

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Example

Given that [math]S_0(x)=\frac{50-x}{50},\quad 0\le x\le 50[/math], we have

[[math]] \begin{aligned} \overset{\circ}e_{x}=\int_{0}^{\infty}{}_tp_x\,dt=\int_{0}^{\infty}\frac{S_0(x+t)}{S_0(x)}\,dt &=\int_{0}^{50-x}\frac{(50-x-t)/50}{(50-x)/50}\,dt \\ &=\int_{0}^{50-x}\frac{50-x-t}{50-x}\,dt \\ &=\int_{0}^{50-x}\left(1-\frac{t}{50-x}\right)\,dt \\ &=\left[t-\frac{t^2}{100-2x}\right]_0^{50-x} \\ &=\left(50-x-\frac{(50-x)^2}{2(50-x)}\right) \\ &=\frac{50-x}{2}. \end{aligned} [[/math]]

In particular, [math]S_0(x+t)[/math] has nonzero value when [math]0\le x+t\le 50[/math], and [math]S_0(x)[/math] has nonzero value when [math]0\le x\le 50[/math]. Also, [math]_t p_x=\frac{S_0(x+t)}{S_0(x)}[/math] holds only when [math]t\ge 0[/math]. It follows that the bounds on [math]x[/math] and [math]t[/math] are given by [math]0\le x\le 50-t[/math] and [math]0\le t\le 50-x[/math].

Assumptions for fractional ages

Previously, we have discussed the continuous random variable [math]T_x[/math] and discrete random variable [math]K_x[/math]. A life table can specify the distribution of [math]K_x[/math] since the values of [math]q_x[/math] for different integer [math]x[/math] can be obtained from the life table. However, the life table is not enough to specify the distribution of [math]T_x[/math], since we do not know the value of [math]q_x[/math] when [math]x[/math] is not an integer. Thus, in order to specify a distribution of [math]T_x[/math] using a life table, we need to make some assumptions about the fractional (non-integer) ages.

In actuarial science, three assumptions are widely used, namely uniform distribution of deaths (UDD) (or linear interpolation), constant force of mortality (or exponential interpolation), and hyperbolic (or Balducci) assumption (or harmonic interpolation). We will define them each using survival functions, as follows:

Definition (Uniform distribution of deaths)

[[math]]S_0(x+t)=(1-t)S_0(x)+tS_0(x+1),\quad 0\le t\le 1.[[/math]]

Definition (Constant force of mortality)

Constant force of mortality assumption assumes

[[math]]\ln S_0(x+t)=(1-t)\ln S_0(x)+t\ln S_0(x+1),\quad 0\le t\le 1.[[/math]]

The equation can be alternatively stated as [math]S_0(x+t)=[S_0(x)]^{1-t}[S_0(x+1)]^t,\quad 0\le t\le 1[/math].

Definition (Balducci assumption)

[[math]]\frac{1}{S_0(x+t)}=\frac{1-t}{S_0(x)}+\frac{t}{S_0(x+1)},\quad 0\le t\le 1.[[/math]]

It is also called hyperbolic assumption

Under UDD assumption, we have some "nice" and simple expressions for various probabilities related to mortality. We can obtain those expressions by substituting [math]S_0(x+t)[/math] by the RHS of the equation mentioned in the assumption.

Example

Under UDD assumption, we have, when [math]0\le t\le 1[/math],

[[math]] _tq_x=1-\frac{S_0(x+t)}{S_0(x)} =1-\frac{(1-t)S_0(x)+tS_0(x+1)}{S_0(x)} =1-(1-t)-t(p_x) =t-t(p_x) =t(1-p_x) =t(q_x). [[/math]]

For the three assumption mentioned, there is a particularly "nice" and simple result for each of them, and we may use those "nice" results for the calculation in practice, rather than applying the definitions. The "nice" result for UDD assumption is mentioned in a previous example: when [math]0\le t\le 1[/math], [math]{}_tq_x=t(q_x)[/math]. The "nice" results for the other two assumptions are as follows:

Theorem (Notable property for constant force assumption)

Under constant force assumption,

[[math]]_tp_x=p_x^t,\quad 0\le t\le 1.[[/math]]

Show Proof

[[math]]_tp_x=\frac{S_0(x+t)}{S_0(x)}=\frac{[S_0(x)]^{1-t}[S_0(x+1)]^{t}}{S_0(x)}=\frac{[S_0(x+1)]^t}{[S_0(x)]^t}=\left(\frac{S_0(x+1)}{S_0(x)}\right)^t=p_x^t.[[/math]]

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Theorem (Notable property for Balducci assumption)

Under Balducci assumption,

[[math]]_{1-t}q_{x+t}=(1-t)q_x,\quad 0\le t\le 1.[[/math]]

Show Proof

[[math]] _{1-t}q_{x+t} =1-\frac{S_0(x+1)}{S_0(x+t)} =1-S_0(x+1)\left(\frac{1-t}{S_0(x)}+\frac{t}{S_0(x+1)}\right) =1-(1-t)\frac{S_0(x+1)}{S_0(x)}-t =1-t-(1-t)p_x =(1-t)(1-p_x) =(1-t)q_x. [[/math]]

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An interesting result under the UDD assumption is related to the independence of two random variables. To simplify the notations, from now on, we let [math]T=T_x[/math] and [math]K=K_x[/math] unless otherwise specified.

Define a continuous random variable [math]S[/math] by [math]T=K+S,\quad S\in (0,1)[/math]. That is, [math]S[/math] is the random variable representing the fractional part of a year lived in the year of death of [math](x)[/math]. For example, if [math]S=0.5[/math], then [math](x)[/math] lives for half year in the year of death.

Then, [math]K[/math] and [math]S[/math] are independent under UDD assumption. This is because

[[math]]\mathbb P(K=k\text{ and }S\le s)=\mathbb P(k\le T\le k+s)={}_{k}p_{x}{}_{s} q_{x+k}={}_kp_x\cdot s\cdot q_{x+k}=(_{k|}q_x)\cdot s=\mathbb P(K=k)\mathbb P(S\le s)[[/math]]

under UDD assumption. Also, we can observe that the cdf of [math]S[/math] is [math]\mathbb P(S\le s)=s=\frac{s-0}{1-0}[/math], which is the cdf of uniform distribution with support [math](0,1)[/math]. This means [math]S[/math] follows the uniform distribution on [math](0,1)[/math] under UDD assumption. Hence, [math]\mathbb E[S]=\frac{1}{2}[/math] and [math]\operatorname{Var}(S)=\frac{1}{12}[/math]. This gives rise to results under UDD assumptions:

• [math]\overset{\circ}{e}_x=\mathbb E[T]=\mathbb E[K+S]=\mathbb E[K]+\mathbb E[S]=e_x+\frac{1}{2}[/math].
• [math]\operatorname{Var}(T)=\operatorname{Var}(T)\overset{\text{ independence }}{=}\operatorname{Var}(K)+\operatorname{Var}(S)=\left(\sum_{k=1}^{\infty}(2k-1){}_k p_x\right)-(e_x)^2+\frac{1}{12}[/math].

These two results give us an alternative way to calculate the mean and variance of [math]T[/math] where only discrete things are used in the calculation. However, we should be careful that these results hold under UDD assumption, so we cannot use these results without UDD assumption.

General References

• Wikibooks contributors. "Fundamental Actuarial Mathematics/Mortality Models". Wikibooks. Wikibooks. Retrieved 14 January 2024.

Wikipedia References

• Wikipedia contributors. "Life table". Wikipedia. Wikipedia. Retrieved 14 January 2024.