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.
There are some constants missing from the integrals, but they cancel out.

If the number of scientist at time $t$ is $y(t) = ae^{bt}$ then the rate of change in the number of scientists is $y'(t) = abe^{bt} = by(t)$ so the rate of change is proportional to the number of scientists.

If you assume stable dynamics, i.e. the rate $z'(t)$ at which new scientists are generated is also proportional to the change in the number of scientists, and the rate at which scientists stop is equal to the rate they were generated $55$ years earlier i.e. $z'(t-55)$ proportional to the number of scientists $55$ years ago, then you get the following results:

$$z'(t)=e^{55b }z'(t-55)$$
$$y'(t) = z'(t)-z'(t-55)=\left(1-e^{-55b }\right)z'(t)$$
$$z'(t)=\frac{ab}{1-e^{-55b }}e^{bt}$$

The number of scientists is the number generated in the last $55$ years is $\int\limits\_{T-55}^{T} z'(t)\,dt$ while the number of scientists ever is $\int\limits\_{-\infty}^{T} z'(t)\,dt$ making the ratio $$\frac{\int\limits\_{T-55}^{T} \frac{ab}{1-e^{-55b }}e^{bt}\,dt}{\int\limits\_{-\infty}^{T} \frac{ab}{1-e^{-55b }}e^{bt}\,dt}$$ and this will produce the same result as the form you are asking about. It will be $\frac{e^{bT}-e^{b(T-55)}}{e^{bT}} =1-e^{-55b}$ and whether this is $0.9$ depends on the value of $b$.

Presumably the assumptions are that there are $10$ times as many scientists now as there were $55$ years ago and that there has always been exponential growth in the number of scientists. I have no idea about the truth of the first of these assumptions, but the second looks implausible to me.