Wikipedia:Reference desk/Archives/Mathematics/2018 January 12

= January 12 =

Life expectancy when you are a baby vs. when you are old
If you are just born into a society where your life expectancy is 80, then someone could say "...your life expectancy is 80..." right?

What if you're, say, 75? You've already survived childhood illness and car crashes in your '20s, right?

So, could someone say "...your life expectancy is now 85..."?*


 * (All things being equal.)

Anna Frodesiak (talk) 01:40, 12 January 2018 (UTC)


 * The third paragraph of our article Life expectancy states, "Mathematically, life expectancy is the mean number of years of life remaining at a given age, assuming age-specific mortality rates remain at their most recently measured levels." According to that definition, your octogenarian has a life expectancy of 5 years, not 85.
 * Check out the last paragraph of Life expectancy, and that paragraph's last reference, Life Expectancy by Age, 1850–2011. A U.S. white male (the first table) born in 2000 had a Life Expectancy at Birth (LEB) of 74.8 years, while an 80-year-old in 2000 had a life expectancy of 7.6 years, though he had an LEB of only 56.34 years (back in 1920, when born). -- ToE 02:53, 12 January 2018 (UTC)
 * Thank you, ToE. Sorry about how I worded my question. Of course, I meant remaining life.


 * If that person you refer to had an LEB of only 56.34 years back in 1920, that was partially because they didn't anticipate a future improvement in healthcare etc., and partially because childhood disease then was more of a consideration, right?


 * My question would be about a baby born today and an 80-year-old today, assuming that nothing on Earth changes in terms of healthcare improvements, etc.


 * So, is the main consideration about the bullets the 80-year-old dodged in the past, and what ones are upcoming? And my guess about that (probably wrong) is that child mortality and wars and such are a lot lower now that before, so the 80-year-old is on track to drop dead within a couple of years if the expected lifespan is 80ish. Is that the way to reckon it? Cheers, Anna Frodesiak (talk) 03:10, 12 January 2018 (UTC)


 * Yes, comparing the octogenarian's future life expectancy of 7.6 years to his LEB of 56.34 years isn't really fair as by definition the values come from age-specific mortality rates at the time of prediction, not taking into account healthcare improvement. Still, back in 1920, an 80-year-old (white male; first table) has a further life expectancy of 5.47 years.
 * So as I understand the way you expressed your main consideration, then no, the 80-year-old today is expected to live longer than an 80-year-old back in 1920. But I think I know what you are getting at.
 * For a given year, look at the Life Expectancy at Birth. Then look at someone who is already at that age and see what their life expectancy is.  And then compare those figures for different years.  For instance, back in 1920 the LEB was 56.34, and a 56.34-year-old had a further life expectancy of between 22.22 and 15.25 years, which we could interpolate to 17.80 years, for an expected age at death of 74.14.  But in 2000 when the LEB was 74.8, a 74.8-year-old had a further life expectancy of between 13.0 and 7.6, which interpolates to 10.4 years, for an expected age at death of 85.2.
 * So in 1920 an LEB=56.34-year-old could expect to live another 17.80 years, while in 2000 an LEB=74.8-year-old could only expect to live another 10.4 years.
 * Is that what you were getting at? -- ToE 03:59, 12 January 2018 (UTC)
 * Yes, totally. Thank you, ToE. I'm really glad I understand this now. I'd bet this doesn't occur to a lot of people who are 80 in a place where that is the expected lifespan. They probably think their time will be up any second. Thank you so much for helping me to understand this. You are very kind. (I'm thinking of England too because that is where I'm from originally.) :) Anna Frodesiak (talk) 08:13, 12 January 2018 (UTC)
 * Someone who's 80, and who thinks their time is up any second because the average life expectancy is 80, is committing the Ecological_fallacy of thinking that an average (life expectancy) applies to an individual. I mention this because one of my statistics classes used that exact scenario to illustrate the fallacy (the example I always gave was "No family actually has 1.5 children") OldTimeNESter (talk) 00:13, 13 January 2018 (UTC)
 * Thank you, OldTimeNESter, for the fine input. I'll read that. Anna Frodesiak (talk) 00:32, 13 January 2018 (UTC)
 * You're quite welcome. Here's a (slightly) risque example of the same principle: https://www.facebook.com/TheEconomist/posts/10155060566504060. OldTimeNESter (talk) 02:32, 13 January 2018 (UTC)
 * I'm afraid I can't read that, OldTimeNESter. My location prohibits it. Anna Frodesiak (talk) 04:59, 13 January 2018 (UTC)
 * Quote: 'He uses the phrase “on average, humans have one testicle” to make the point that the mean can be a misleading description of a population'. Double sharp (talk) 11:48, 13 January 2018 (UTC)
 * I'm head of state of 0.00000003 countries and have died 0.9 times. Sagittarian Milky Way (talk) 18:20, 13 January 2018 (UTC)


 * That Facebook page merely links (with a quick summary) to this article: —Tamfang (talk) 18:41, 13 January 2018 (UTC)
 * Thank you, Tamfang. I'll read that. Anna Frodesiak (talk) 21:14, 13 January 2018 (UTC)

The probability that the dying age X of a random person is x, is
 * $$P(X=x)$$

The expected dying age of this person is
 * $$\sum_{x=0}^\infty x P(X=x)$$

The conditional probability that the dying age X of a random person is x, when he is not yet dead at age 75, is
 * $$P(X=x|x>75)$$

The expected dying age of this person is
 * $$\sum_{x=76}^\infty x P(X=x|x>75)$$

Bo Jacoby (talk) 19:04, 14 January 2018 (UTC).

It's also interesting to consider the time dependent probability per unit time $$P(x,t)$$ of someone aged $$x$$ at time $$t$$ to die right at that time. If $$P(x,t)$$decreases sufficiently rapidly as a function of time, then the expected age of dying can be infinite. So, even though at no time the chance of dying becomes exactly zero at any age, it's still possible that you can expect to live till eternity. Count Iblis (talk) 19:34, 14 January 2018 (UTC)