## On Celebrating Birthdays

There could be many reasons to celebrate a birthday:

• came a year closer to adulthood
• survived one more year
• chance to party!
• chance to get b’day bumps without fear of being labeled a masochist

But several months ago, an old person remarked to me, “I hate celebrating my birthday because it means being a year closer to the end”. That reasoning looked irrefutable. I mumbled something about surviving another year, and slunk away. Although this happened several months ago, the thought stayed with me. I felt there had to be something wrong with this argument. And then, a few days ago, I realized how a refutation was possible. That is what I want to cover in this article.

“A year gone, hence a year closer to the end” — is based on the assumption that the end is fixed, that the moment of death is destined, frozen in advance. But in fact, the future is not frozen — it holds many possibilities. For example, one may become a centenarian, or one may die in a freak accident tomorrow. However, one can use statistics to estimate one’s chances of survival. Now, it turns out that by following this statistical reasoning, one can show that “a year closer to the end” is wrong thinking. However, before I develop this argument properly, we must take a detour into some necessary concepts: lifespan, life expectancy, mortality rate, and life table.

Lifespan is simply the number of years a person lives. Obviously, it can only be determined post facto, after he or she is gone.

Life Expectancy is the average lifespan of a population, again measured post facto, after every single individual is gone.  Note that a “population” may be chosen in different ways — e.g. all individuals in a state or country, or only the females in a region, or only females within a certain income range, or only Japanese females within a certain income range, and so on. Once such numbers are collected for a large sample size, interesting patterns emerge. For one thing, one can determine how many people died between age n and n+1. This number, generally expressed as a count per 100,000, is called mortality rate.

It turns out that mortality rate is not constant. See the chart of age versus mortality rate below (males, USA, 2010)  [9]. There is a blip of high child mortality, then a long tail that builds into a tall peak in ripe old age. The area under the bars equals 100,000. Notice that had the mortality rate been constant, the curve would have been flat at height ~1000 (assuming max age of 100 years).

C(n) = Conditional life expectancy at age n is the average lifespan of a population that has already survived to age n. Clearly, life expectancy = C(0). In the chart, C(0) is 79. Visualize it as the point along the horizontal axis where the blue shape will balance on a knife edge.

Now we come to something interesting: how does one calculate C(n) for various values of n? Notice that with increasing values of n, one discards the individuals that did not survive to age n. That is, the blue bars to the left of n are to be discarded. It is obvious that the knife edge has to be shifted to the right to keep the truncated shape balanced. That is, C(n) keeps increasing with n.

One can also understand why C(n) rises by using an analogy. Consider a pile of pebbles of various sizes. Each pebble represents a person, and the pebble’s weight represents the person’s lifespan. Say, a 5 gram pebble corresponds to a lifespan of 5 years, a 100 gram pebble corresponds to 100 years. Of the initial pile, the average pebble weight represents life expectancy. Now, to find C(n), we remove all pebbles weighing less than n grams. Then C(n) is the average weight of the remaining pile. It should be obvious that as we keep increasing n, that is, keep throwing out the lighter pebbles, the average weight keeps on increasing.

Life table is simply a table of C(n) for different values of n. An abbreviated table adapted from [4] is given below (note: sampled population is different than that of the previous chart).

Table: Conditional Life Expectancy C(n) versus age

 AGE C(n) AGE C(n) AGE C(n) AGE C(n) 0 76.33 1 76.81 26 77.58 51 79.79 76 86.51 2 76.84 27 77.65 52 79.94 77 86.93 3 76.86 28 77.72 53 80.11 78 87.36 4 76.88 29 77.79 54 80.29 79 87.81 5 76.89 30 77.86 55 80.47 80 88.28 6 76.90 31 77.93 56 80.67 81 88.76 7 76.91 32 78.00 57 80.87 82 89.27 8 76.92 33 78.07 58 81.09 83 89.80 9 76.93 34 78.15 59 81.32 84 90.34 10 76.94 35 78.22 60 81.55 85 90.91 11 76.94 36 78.29 61 81.79 86 91.50 12 76.95 37 78.37 62 82.04 87 92.11 13 76.96 38 78.44 63 82.30 88 92.74 14 76.97 39 78.52 64 82.57 89 93.40 15 76.99 40 78.60 65 82.84 90 94.08 16 77.02 41 78.68 66 83.12 91 94.79 17 77.05 42 78.76 67 83.40 92 95.52 18 77.08 43 78.85 68 83.70 93 96.27 19 77.13 44 78.95 69 84.01 94 97.05 20 77.18 45 79.04 70 84.32 95 97.85 21 77.24 46 79.15 71 84.66 96 98.68 22 77.30 47 79.26 72 85.00 97 99.53 23 77.37 48 79.38 73 85.36 98 100.39 24 77.44 49 79.51 74 85.73 99 101.27 25 77.51 50 79.64 75 86.11 100 102.15

A full plot of C(n) vs n is given below. It is obvious that life expectancy rises with age, especially after age 60.

Conditional Life Expectancy C(n) vs age

So now it is time to refute “passing of one year brings one a year closer to the end”. Why is this incorrect? First of all, the future is not fixed and there is no fixed predestined lifespan for a particular person. Rather we only have statistical estimates, which change as new information comes to light. Thus the very fact that one has survived another year is new information that pushes out the endpoint. This yearly push amounts to a substantial fraction of a year for super senior citizens!

So one can celebrate a birthday not only to celebrate having survived another year, but also for having pushed out one’s life expectancy by the very fact of having survived another year.

[1] Life expectancy https://en.wikipedia.org/wiki/Life_expectancy

[2] Life table https://en.wikipedia.org/wiki/Life_table

[3] Mortality rate https://en.wikipedia.org/wiki/Mortality_rate

[4] Actuarial Life Table https://www.ssa.gov/oact/STATS/table4c6.html

[5] Mortality Database http://www.who.int/healthinfo/statistics/mortality_rawdata/en/

[6] country life tables http://apps.who.int/gho/data/?theme=main&vid=60740

[7] World life table http://www.worldlifeexpectancy.com/world-life-table

[8] Health Inequality Project https://healthinequality.org/data/