## Bleg: What’s the most recent day no one alive was born

Inspired by Michael Lugo’s post on reconstructing a person from their DOB, zipcode, and gender.

If you, for whatever reason, ever watch the Today show, you’ll notice that one of the recurring features is the hosts listing the names of some men and women who are turning 100. Becoming a centenarian is a reasonably big accomplishment — in the U.S., it nets you a congratulatory letter from the President, for example. But if you look into it, you’ll notice that you can find someone turning 100 on pretty much any given day. Usually not someone particularly well-known, but certainly someone. (I tried to find someone famous and vaguely math-related who just turned or is turning 100 for this post, but couldn’t; however, the fascinating economist Ronald Coase turned 99 last week.) It’s almost certainly true that on any given day, someone somewhere in the world is in fact celebrating their 100th birthday. But go ten years further, and you find almost no one who lives to 110. Actually, I know of only one supercentenarian, living or not, who is interesting for reasons apart from his longevity — the late Vietoris, the topologist, probably best known as half of the Vietoris-Rips complex and the Mayer-Vietoris sequence. Odds are pretty good that no one alive is turning 110 today, or tomorrow, or (sadly) New Years’ Day.

So… a question is starting to take shape. On every day between December 29, 1909, and today, someone was born who is still living today. But much earlier than that, and the above statement begins to be false. So what’s the most recent day that no one living was born on?

Unfortunately the question seems impossible to answer precisely — even today in some countries there’s no reliable system to record births and demographic data, and 100 years ago the situation was far worse. But we can certainly try to make an educated guess, or at least think about how we’d go about trying to make an educated guess!

So we’ll start off with our simplifying assumptions. First of all, in the analysis we’re going to have to consider the probability that a person who’s lived to N days will live to N+1 days. (Or a coarser version of this statistic.) Obviously this probability is different for each person — a 104-year-old in Bangladesh with a terminal disease and without access to good medical care has a much shorter life expectancy than a healthy person of the same age in, say, New Jersey. But this complicates matters hugely, so we’ll say that for each person the probability is the same.

In addition, we’ll assume that the birth rate was constant, say, 1900 and 1910, and that someone born in 1902 had the same probability of living to 100 and someone born in 1909. Again, these are simplifying assumptions.

So once you’ve made these assumptions, you end up with a sequence of random variables $X_N$ that describe how many people who have lived exactly N days are still living. Under the above (unrealistic) assumptions, $X_N$ is approximately a binomial distribution, with the probability decreasing exponentially with N.

So the first estimate we can make is to see when the expected value $E(X_N) < 1$. But this isn’t all that good — probably the first N with $X_N = 0$ came way earlier — and so a better thing to do is to estimate

$Pr[\bigcup_{j \leq N} X_j = 0] \leq \sum_{j \leq N} \text{Pr}[X_j = 0]$.

But, when the $X_j$ are binomial, this probability is easy — it’s just $1 - p^{jM}$ for some fixed probability p and integer M. So the sum is

$N + 1 - (1 + p^M+ \ldots + p^{NM}) = N + 1 - \frac{1 - p^{MN + M}}{1 - p^M}$.

As long as this is small, the probability  is high that none of the $X_j$ are 0 for $j \leq N$.

Can we do better than this? What if we replaced the above model by something more realistic, where the death rate increases as N increases?

What’s (approximately) the most recent day no one alive was born?

### 2 Responses to “Bleg: What’s the most recent day no one alive was born”

1. Michael Lugo Says:

Here’s a thought on how to go from data at the level of years (which might exist somewhere out there although I’m having trouble finding it) to a decent guess at the level of days. Let’s say we know that N people were born in 1905 and are still alive. Then the number of people born on any given day in 1905 is approximately Poisson with mean N/365;. The probability that, for any given day in 1905, at least one person born that day is still alive, is then about exp(-N/365). Assuming that the days are independent, the probability that at least one person born on every day in 1905 is still alive about (1-exp(-N/365))365. Of course one could chain this together for various years to get some sort of estimate.

I think the Census has data of the type I already described, although I’m having trouble finding it. Apparently demographers refer to a chart showing the number of people of each age as a “population pyramid”, and somewhat annoyingly most of the ones I can find lump everyone over 85 together. This table might be useful, although it uses what seems to be a standard actuarial fiction: it’s a “life table” which assumes that people of every age have the same age-adjusted mortality rates that they did in 2005, which is not exactly what we need to answer this question.

And of course that’s only for the United States. A small change in the average life expectancy between countries could show up as a large difference in the number of people who survive to extreme ages.

Without the data, it looks like there are various theoretical models of longevity that demographers use, although I’m not sure how accurate any of those are. Rather understandably, it looks like demographers are more interested in having models that work for most individuals than in having models that work in the tails of the age distribution.

2. TonyK Says:

You’d better be more specific if you want your question to even have an answer. What do you mean by ‘day’? The simplest solution is to designate a timezone, but which one do you pick? Another solution is to accept any 24-hour period, but I don’t think your question can be interpreted in this way.