Question: I am reading Richard Hamming's book and came across a back of the envelope calculation I am having trouble reconciling. The relevant assumptions in the calculation are:

1. assume the number of scientists at any time `t` can be given by:
$$
y(t) = \mathrm{ae}^{bt}
$$
2. 90% of scientists who have ever lived are alive today (assume a 55 year life span for a scientist). This is modeled as:
$$
0.9 = \frac{\int\_{T-55}^T \mathrm{ae}^{bt}\, dt}{\int\_{-\infty}^T \mathrm{ae}^{bt}\, dt}
$$

I cannot wrap my head around the numerator here. Given #1, the numerator instead seems to denote the cumulative number of scientists *who have been alive during the past 55 years*, rather than the total number of scientists *alive today*. Some (most) scientists who were alive 55 years are probably no longer alive today, so it seems to me that the numerator actually inflates the count of scientists alive today.

Am I missing something here? And as a follow up, how would you model the total number of scientists alive today, given the information in the assumptions?

Answer: 
There is something strange in the given calculation. If $y(t)$ represent the total number of scientists at time $t$ (alive or dead) then the number of scientists at time $T$ is $y(T)$. However, because scientists live just $55$ years then many of the scientists $y(T)$ are already dead, indeed $y(T-55)$ are all dead at time $T$, so the total number of scientists alive at time $T$ will be $y(T)-y(T-55)$, thus the proportion is just
$$
\frac{y(T)-y(T-55)}{y(T)}\tag1
$$
Maybe the wording of the model is bad, and trying to follow the given calculations then it seems that the population of scientists is modelled by a strange exponential distribution who support is $\mathbb{R}$ instead of just $[0,\infty )$. Under this assumption the total number of scientists, alive or not, at time $T$ is given by $\int\_{-\infty }^T y(t) \mathop{}\!d t$, however the alive scientists are just who had birth in the period $[T-55,T]$, or in other words

$$
\overbrace{\int\_{-\infty }^Ty(t) \mathop{}\!d t}^{\text{total scientists at time } T}-\overbrace{\int\_{-\infty }^{T-55}y(t) \mathop{}\!d t}^{\text{total scientists dead at time } T}=\overbrace{\int\_{T-55}^T y(t)\mathop{}\!d t}^{\text{ living scientists at time }T}\tag2
$$

Therefore the total number of living scientists at time $T$ is $\int\_{T-55}^T y(t) \mathop{}\!d t$, hence the given result follows.

But note that in this last model $y(t)$ **is not** the number of scientists at time $t$, it is just the density function of the model of the total amount of scientists.
The calculation given in your book, taken literally, overcounts in both the numerator and the denominator. Since the integral of the exponential is a very similar exponential, however, it is somewhat harmless, and I somewhat wonder if $y(t)$ is instead supposed to be a birth rate of new scientists instead of the total count.

The calculation more precisely should be
$$\frac{y(T)}{\sum\_{k=0}^\infty y(T-55k)}$$

Since $y(t)$ is increasing, however, we can estimate [with an associated error] both the numerator and denominator by the integrals given in your book.

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As a mostly unrelated note, population growth shouldn't be modeled by the exponential model for long time scales, so if I were to complain about anything, it would be using an inappropriate model.