In the answer to a recent question (ID 187759), it says that the total
US death rate is 848.9 per 100,000 (call this fraction A).  I assume
this means that in any given year, the expectation value of the number
of deaths in a random sample of 100,000 people is 848.9 (meaning one
of every 117.8 are expected to die that year).  I have seen numbers
similar to this before.  Meanwhile, the average age of death is 76.9
or so (http://www.cdc.gov/nchs/fastats/lifexpec.htm).  My question is:
how can both these be true?

To illustrate my confusion, here are a couple of examples: 

If everyone lived to exactly the same age, so that the death rate at
that age is one, and at all other ages zero, the age would be have to
be 1/A, or about 117.8 years, to satisfy the total death rate
statistic above.

If death rate were independent of age, so that the probability of
dying in a given period of time is the same for everyone, regardless
of age, the distribution of age of death would be exponential, and the
average lifetime would be -(ln(1 - A))^-1, or about 117.3 years.

Neither of these cases is realistic: in the first, death rate as a
function of age is a single spike, and in the second, it is uniform;
in reality, the death rate is nonzero everywhere, but increases with
age.  However, this seems like sort of a middle ground between my two
idealized cases, both of which give approximately equal, rediculously
long, life expectancies.  What is it about the distribution of death
rate with age in the real world that allows average lifetime to be so
much shorter, or have I misunderstood something?

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Clarification of Question by racecar-ga on 09 Apr 2003 11:04 PDT

Maybe it's that the population is not in steady state--there were more
births in the last few decades than in the decades before so death
rate is low, and once population growth stops, death rate will go up?

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Clarification of Question by racecar-ga on 09 Apr 2003 12:19 PDT

Ok, I think I'm an idiot, and that in a steady-state population where
all members are statistically the same, the life expectancy is just
the inverse of the death rate.  The life expectancy for the uniform
death rate case is not -(ln(1 - A))^-1 as I wrote above, but just 1/A.
 The fact that the life expectancy in the US is not the inverse of the
death rate is just a reflection of the fact that we are not in steady
state.  Anyone who can confirm this is still welcome to the question
fee.

Actuarial science is a wonderfully subtle application of probability.

In reality the probability of death is relatively high both in infancy
and in "old age".

If you are interested I can scrounge up some actuarial tables for you,
but since the percentage of population over 100 is so small, actuaries
generally just assume a constant death rate over a certain age.

If you work with a fixed population, like Civil War veterans, you can
get some surprisingly accurate numbers for the declining size of the
population with the passage of time.  As you point out, in a dynamic
population that is non-homogeneous because of birth rates,
immigration, war, epidemics, etc., accurate predictions are more
difficult to come by.

regards, mathtalk-ga

I had similar puzzlement, and was going to ask some friends who are
better at math than I. I tentitively assumed it was similar to the
descrepency that occurs in compound interest. I can see where the
imigrants, those who leave the country and increasing population squew
somewhat the numbers, but the amount seems excessive. Perhaps you can
get some figures for West Germany which I believe had a decreasing
population for the decade before the Berlin wall came down. Here the
discrepency should be in the opposite direction if we have found the
real cause.
 My friend thinks it is because senior citizens are living longer now
even if younger people have the same mortality as a decade or two ago.
 He is going to ask his sister who was the supervisor for the actuary
department of a life insurance company.  Neil