zachisbest
zachisbest
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July 15th, 2010 at 1:25:04 AM permalink
Life expectancy for people of various ages has been calculated and summarized with data on the following chart:


However, I want to know the life expectancy of a population of people. [i.e. when will they all be dead]

Say I have two people: a thirty-year-old male [me] and a twenty-eight-year-old female [my gf]. According to the chart I will live another 46.89 years and she will live another 53.22 years.

But...how long is it expected until we both are dead? [It can't just be 53.22 years]

How do I calculate this?

What about if we live on a commune with 30 30-year-olds... how long until the last of us dies?
dwheatley
dwheatley
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July 15th, 2010 at 5:05:36 AM permalink
Not an easy question. If the time until you die: X1 is a random variable, the actuarial life tables will give you its distribution. The time until your gf dies: X2 is also a random variable, with a different distribution.

You want E[ max(X1,X2) ]

In the general case, where you live in a commune with n people, you want E[ max(X1,X2,...Xn) ].

This is called the nth order statistic of a population of size n. Order statistics are complicated, especially when the distributions of the random variables are not identical. http://en.wikipedia.org/wiki/Order_statistic

If you could assume that you actually all had the same life expectancy (the r.v.s X1,...Xn are iid), there is a closed form for the distribution of the max in terms of the original distribution of X1 which you could use to calculate the expectation. But, that's not the case...

If you actually want a number, you can crunch out all the joint probabilities of you and your gf living to certain ages in Excel (or similar program), and calculate the expectation of the oldest person. Doing this for any more than 2 people would be ugly.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
dwheatley
dwheatley
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July 15th, 2010 at 5:16:50 AM permalink
For a 30m and 28f, i get 86.09 as the expected age of the oldest person. This is slightly different then the life expectancy of the person who dies last, which I get as 57.34 years.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
DJTeddyBear
DJTeddyBear
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July 15th, 2010 at 5:25:56 AM permalink
Very interesting question.

I have no answer, but a comment...


I once heard a statement that people whose spouse is still alive tend to live longer.

After giving that some thought, I came up with TWO very obvious reasons for this.

1 - Once a spouse dies, the survivor tends to lose the will to live.

2 - Since it is unlikely that a couple will die together, the fact that both are alive, raises the odds that at least one will be still alive in the forseeable futre remains high.
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
dwheatley
dwheatley
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July 15th, 2010 at 5:50:55 AM permalink
DJ, I agree with point (1), I saw it myself with my grandma. She's still living a good life, but almost didn't after a heart surgery she had to go through on her own.

Point (2) I am not so sure about. First, maybe couples die together more often than 2 strangers. Accidental deaths (car crashes, fires, etc.), while being fairly rare, could easily take a couple. Second, how does being a couple affect this statement? Can't I take two strangers and make the same claim?
Wisdom is the quality that keeps you out of situations where you would otherwise need it
DJTeddyBear
DJTeddyBear
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July 15th, 2010 at 7:26:05 AM permalink
Quote: dwheatley

Can't I take two strangers and make the same claim?

Actually, yeah.

To put this in math terms, the longer two events remain true, the bigger the chances that at least one of them is true.


But to be honest, I'm not sure if it really translates to math all that easily.


And I agree that the chance of a couple dying together, although small, is atronomically larger than the chance of two random people that you're merely monitoring, dying at the same time. But does it really chance things much?
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
Wizard
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Wizard
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July 15th, 2010 at 7:51:13 AM permalink
First, it would be appropriate to use cohort life tables, as opposed to the period life table you linked to. I tried to find cohort life tables online, but was unsuccessful. However, we can still use the table provided. It may underestimate how long you will live slightly, because it won't take into account future increases in life expectancy.

Answering your question involved creating a large matrix of the probability of each combination of year of death for you and the 28-year-old female. Forgive me if I don't get into detail. The bottom line is that I show that first one of you will die in 41.8 years, and the latter death will be in 57.3 years. Both figures round down, in other words, you don't get credit for partial years.

p.s. In the 90's I worked in the office that created that table.
It's not whether you win or lose; it's whether or not you had a good bet.
konceptum
konceptum
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July 15th, 2010 at 8:28:10 AM permalink
I prefer to use my own actuarial table which assumes that I will die tomorrow. Thus I have lots and lots of fun every day of my life, because I never know if it's going to be my last one.

For some reason, I've racked up a lot of debt.....
DJTeddyBear
DJTeddyBear
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July 15th, 2010 at 8:59:16 AM permalink
Quote: konceptum

...For some reason, I've racked up a lot of debt.....

LOL!

Did you at least hedge the bet and get Life Insurance?
Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
odiousgambit
odiousgambit
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July 15th, 2010 at 12:31:43 PM permalink
Quote: Wizard

... it won't take into account future increases in life expectancy.



are further increases still a given? The News would have you believe that with the obesity and diabetes epidemic they report, life expectancy could be going down.
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell! She is, after all, stone deaf. ... Arnold Snyder

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