## Different types of survival curves : longevity and life expectancy

You have done an exploratory experiment using ExpectedLifeTime.nb in class last Saturday. You should have seen that the longevity was Type I (60.43), Type II (21.72), and Type III (19.28), in that order.

More interesting than the longevity comparison is comparison of the relationship between the longevity and the length of remaining life at other ages between each type.

I didn't explain how to get life expectancy at an arbitrary age ${x}^{\ast }$$x^*$ in the class. The life expectancy of an individual at ${x}^{\ast }$$x^*$-age is

$\sum _{x={x}^{\ast }}^{\mathrm{\infty }}\frac{{l}_{x}}{{l}_{{x}^{\ast }}}=\frac{1}{{l}_{{x}^{\ast }}}\sum _{x={x}^{\ast }}^{\mathrm{\infty }}{l}_{x}$

or, if you can define $l\left(x\right)$$l(x)$ as any continuous function,

${\int }_{{x}^{\ast }}^{\mathrm{\infty }}\frac{l\left(x\right)}{l\left({x}^{\ast }\right)}dx=\frac{1}{l\left({x}^{\ast }\right)}{\int }_{{x}^{\ast }}^{\mathrm{\infty }}l\left(x\right)dx$

"$\mathrm{\infty }$$\infty$" in eq. is for mathematical convenience. Realistically, this is the age at which they will live the longest.

Think about why the equations look like this form. I hope you are convinced.

Let's look at longevity and life expectancy, for exammple, at age-20 (${x}^{\ast }=20$$x^{*}=20$ in the above eq.).

In Type I, a newborn has a longer remaining life than a 20-year-old adult. In Type II, the remaining life for a newborn and for an individual of age-20 is the same. In Type III, a age-20 adult has a longer remaining life than a newborn.

At first, the pattern for type 1 seems to follow our intuition, and we may wonder about the pattern for type II and type III. Perhaps, with a little thought, you will be able to understand why this happens.

Even so, let's examine why this is so below. The table shows some of the yearly values of the ${l}_{x}$$l_x$ for each type.

file download $\phantom{\rule{0.278em}{0ex}}⟸\phantom{\rule{0.278em}{0ex}}$$\color{red}\LARGE\impliedby$ Click ! You can get the file, in which ${l}_{x}$$l_x$ numerical data are included.

We can derive other interesting parameters from ${l}_{x}$$l_x$.

${l}_{x}$$l_x$ is the probability that a newborn is alive at age-$x$$x$.

${s}_{x}=\frac{{l}_{x+1}}{{l}_{x}}$$s_x=\frac{l_{x+1}}{l_{x}}$ is the probability that an individual of age $x$$x$ survives to age $x+1$$x+1$.

${d}_{x}=1-{s}_{x}$$d_x=1-s_x$ is the probability that an individual of age $x$$x$ die by age $x+1$$x+1$. ${d}_{x}$$d_x$ is the complement of ${s}_{x}$$s_x$.

Examples for type I case

${l}_{10}=0.9994$$l_{10}=0.9994$ means that a newborn has a $99.94\mathrm{%}$$99.94\%$ chance of survival at age-10.

${s}_{10}=\frac{{l}_{11}}{{l}_{10}}=0.9997$$s_{10}=\frac{l_{11}}{l_{10}}=0.9997$ means that an age-10 individual has a $99.97\mathrm{%}$$99.97\%$ chance of survival at age-11.

${d}_{10}=0.0002349$$d_{10}=0.0002349$ means that an age-10 individual has 0.023...% chance of dying before reaching age-11.

Comparing ${s}_{x}$$s_x$ (or ${d}_{x}$$d_x$) along $x$$x$ across the types, we can understand the reason for the different patterns in the comparison of longevity and life expectancy across the types.