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13 Exercises: Accumulation

Exercise 13. 1 One measure of the amount of transportation (for instance, delivered by a bus system) is person-miles (generally written “passenger-miles”).

Consider a simple instance of an intercity bus that takes on and drops off passengers in going from city A to B to C to D. The distance from A to B is 100 mi, from B to C is 50 mi, and from C to D is 200 mi.

The bus starts from city A, it is carrying 10 passengers. At city B, four passengers get off and 11 get on. At city C, 7 passengers get off and 14 get on. All passengers disembark at city D.

How how much transportation service has the bus provided?

Exercise 13. 2 EXERCISE (for integration): Referring to the lognormal distribution in “Uncertainty” chapter, find out what fraction of the income goes to the top 10% or 1% of income earners, modeling income as a lognormal distribution.

Exercise 13. 3 The graphic in Fig E13. 1 appeared in an editorial in the New York Times. The editorial states this fact: “More than 400,000 Americans died from drug overdoses between 2020 and 2023.”

Assume that the graphic shows a continuous function of the date.

Would the graphic, as labeled, be consistent with the statement?
What are the units for the vertical axis if the graphic is to be consistent with the statement and show a continuous function of the date?

Figure E13. 1: Overdose deaths in the US. (Source: New York Times, August 28, 2025)

Exercise 13. 4 (Example: Predicting the US population) Every 10 years, starting in 1790, the US Census Bureau carries out a constitutionally mandated census: a count of the current population. The overall count as a function of year is shown in UNRESOLVED fig-pop-graph. [Source]

In the 230 years spanned by the census data, the US population has grown 100-fold, from about 4 million in 1790 to about 330,000,000 in 2020.

It is tempting to look for simple patterns in such data. Perhaps the US population has been growing exponentially. A semi-log plot of the same data suggests that the growth is only very roughly exponential. A truly exponential process would present as a curve with a constant derivative, but the derivative of the function in the graph is decreasing over the centuries.

Insofar as the slope over the semi-log graph is informative, it amounts to this quantity: \[\partial_t \ln(\text{pop}(t)) = \frac{\partial_t\, \text{pop}(t)}{\text{pop}}\] This is the per-capita rate of growth, that is, the rate of change in the population divided by the population. Conventionally, this fraction is presented as a percentage: percentage growth in the population per year, as in Fig E13. 3.

Figure E13. 3: Annual per capita growth rate of the US population (percent)

The dots in the graph are a direct calculation from the census data. There is a lot of fluctuation, but an overall trend stands out: the population growth rate has been declining since the mid-to late 1800s. The deviations from the trend are telling and correspond to historical events. There is a relatively low growth rate seen from 1860 to 1870: that is the effect of the US Civil War. The Great depression is seen in the very low growth from 1930 to 1940. Baby Boom: look at the growth from 1950-1960. The bump from 1990 to 2000? Not coincidentally, the 1990 Immigration Act substantially increased the yearly rate of immigration.

If the trend in the growth rate continues, the US will reach zero net growth about 2070, then continue with negative growth. Of course, negative growth is just decline. A simple prediction from ?@fig-pop-growth is that the argmax of the US population—that is, the year that the growth rate reaches zero—will occur around 2070.

How large will the population be when it reaches its maximum?

In Block 2, we dealt with situations where we know the function $f(t)$ and want to find the rate of change $\partial_t f(t)$. Here, we know the rate of change of the population and we need to figure out the population itself, in other words to figure out from a known $\partial_t f(t)$ what is the unknown function $f(t)$.

The process of figuring out $f(t) \longrightarrow \partial_t f(t)$ is, of course, called differentiation. The opposite process, $\partial_t f(t) \longrightarrow f(t)$ is called anti-differentiation.

In this block we will explore the methods for calculating anti-derivatives and some of the settings in which anti-derivative problems arrive.

Theorem 1 Population growth

The predictions from the accumulate-population-growth model are shown as a magenta line in Fig E13. 4.

Figure E13. 4: Predicted US population based on the historical linear decline in per-capita growth.

According to the accumulation model, the population peaks in 2075 at 390 million. We will be back down to the present population level in about 100 years.

Professional demographers make much more sophisticated models using detailed data from many sources. The demographers at the US Census Bureau predict that the population will reach a maximum of 404 million in 2060, shown by the little blue dot in Fig E13. 4. that is not too different from what we got by analyzing just the raw census numbers.

Exercise 13. 5 Look at exponential growth, and at the rate of change of exponential growth which itself grows exponentially

Exercise 13. 6 Look at exponential decay—growth reversed in time. This won’t run in to the limits of exponential growth since its getting closer and closer to zero.

Exercise 13. 7 Use US Population data to plot rate of change versus population. This is the same format as the dynamical rule. For exponential growth, the pattern should be steady.

Exercise 13. 8 The New York Times Magazine (Nov. 30, 2025) reported on a surge in e-bike injuries. The article sensibly pointed to the rapid increase in e-bikes sales as a factor in the surge.

As the pandemic continued, the number of e-bike accidents increased. “You would expect that,” Alfrey says, “because sales were skyrocketing.” Indeed, in 2022, over a million e-bikes were sold in the United States, up from 287,000 in 2019, according to the Light Electric Vehicle Association.

The article pointed out that the increase in e-bike sales has not been as sharp as the increase in injuries.

During the same [three]-year¹ period when nationwide sales quadrupled, e-bike injuries increased by a factor of 10, to 23,493 from 2,215, according to the National Electronic Injury Surveillance System. A study by the University of California, San Francisco, found that from 2017 to 2022, head injuries from e-bike accidents increased 49-fold.

What do you think the units are for the reported e-bike sales?

What do you think the units are for the reported injuries?

What’s your opinion? Is it appropriate to compare the rapidly growing e-bike yearly sales to yearly injuries.

What is the approximate doubling time for e-bike sales, according to the information presented in the article?

Explain the reasoning that led to your conclusion about the doubling time.

Accident rates are typically proportional to exposure; the more an activity is undertaken, the larger the number of accidents per year associated with that activity. Which of these would you expect to be proportional to the number of accidents per year?

whk-92-5

Bicycle sales per year is a rate of change: the change in the number of bicycles divided by the change in time. We are given information about bicycle sales, but we need information about the total number of bikes on the road. Making this conversion in the form of information—from a rate of change of a quantity as opposed to to the quantity itself—can be accomplished by accumulating the rate of change over time.

To perform the accumulation, we need to look at the sales in each year of the period. We know the number of sales in 2019 and 2022: about 250,000 and one-million respectively.

Using the numbers in the previous paragraph, how many doubling occurred from 2019 to 2022? How many years elapsed from 2019 to 2022?

whk-92-6

The doubling time is the span of years divided by the number of doublings. If D is the doubling time in years, then the proportional increase from year to year is double(1 / D).

We’ve done the calculation for you about bicycles sales in each year using the proportional increase from year to year. Your job is to use the accumulation method to calculate the number of bikes on the road in each year.

Year	2013	2014	2015	2016	2017	2018	2019	2020	2021	2022
Sales	15K	25K	40K	60K	100K	160K	250K	400K	630K	1000K
On road	15K	40K	80K	…	…	…	…	…	…	…

Assuming that the number of bikes on the road in any year consists of the bikes sold through that year, how many bikes were on the road in 2017?

Using the same assumption as in (7), how many bikes were on the road by 2019?

Using the same assumption as in (7), how many bikes were on the road by 2022?

The article says that e-bike injuries increase 10-fold over the period from 2019 to 2022, while bike sales only increased 4-fold. Coincidentally, the number of bikes on the road increased by slightly more than 4-fold from 2019 to 2022, so the 10-fold increase in injuries does reflect a growing injury rate per bicycle.

The number of head injuries, in contrast, increased 49-fold from 2017 to 2022. But there was a 7-fold increase in the number of bicycles on the road. So the head injury rate per bicycle increased “only” 7 fold.

Exercise 13. 9 Simple accumulation: growth dynamics: one variable, logistic, sunken gaussian (difficulty finding mate, …).

Exercise 13. 10 Let’s pick up on the graphic in Exercise 3, repeated below.

A reasonable form for the data represented by the graphic would be a data frame like this, where the date column is sorted in ascending order.

row	date	year	incident_file
1	2000-01-01	2000.00274	nxg93c.pdf
2	2000-01-01	2000.00274	pzd61f.pdf
3	2000-01-02	2000.00548	ytr41u.pdf
…	…	…	…

Assume that each incident file pertains to a single overdose death. The actual data frame would be much longer and would have scores of entries for each date.

The New York Times editorial in which the graphic appeared gave this fact: “More than 400,000 Americans died from drug overdoses between 2020 and 2023.” Based on this statement, roughly how many Americans died during the complete period included in the graph? Explain your reasoning.
Assuming the data frame covers the entire period 2000-2024, roughly how many rows would be included?
Suppose you constructed, by interpolation, a continuous function for cumulative deaths (starting in 2020) at any instant from the data frame. What variables would be needed as inputs to the interpolation operation?
How can you tell from the graphic that it does not display the cumulative death function?
What mathematical operation would you perform on the cumulative death function

Exercise 13. 11 (Example) SOME MISCELLANEOUS IDEAS FOR EXERCISES

Likelihood for observing a car going 25,000 miles without an accident.

How is it that we could calculate the proper balance between Natal and Cancer by looking just at the derivative function, QALYs per dollar, and not the QALY vs cost function itself.

Example

Looking at the mean time between events for the two hypotheses in Chap 11 on the interval between events.

Tie to uncertainty chapter

Referring to Figure 7.1, explain how the cumulative distribution, with the output rescaled to 0 to 1, provides the official definition of the 95% interval.

Exercise 13. 12 In the new-housing example in the text, a claim was made that the number of new households increases by about 1% a year. Fig E13. 5 shows the data on which this claim was made.

Figure E13. 5: Number of US households versus time. Source

Do the calculation of household growth yourself, based on these data. Explain your calculation and whether the result agrees with the claim in the text.

Exercise 13. 13 The US government debt (usually called the “National Debt”) is approximately $40 trillion as of early 2026. Written out, that’s $40,000,000,000,000. Or, $4e13.

The “gross domestic product” (GDP) of the US is about $30 trillion per year. The per-year is important. It’s the same as $2.5 trillion per month, or $1 million per second, or $3000 trillion per century.

Economic journalists often describe the debt condition of a country as ratio of debt to GDP. For instance, looking at the amounts $40 and $30 trillion, they would measure the US debt is 125% of GDP. A journalist’s rule of thumb is that ratios above 100% are a sign of trouble.

But, since GDP can also be measured as $3000 trillion per century, wouldn’t it be right to say that the ratio of debt to GDP is 1%? Explain why or why not the 125% figure is meaningful.

Tip

Comparing GDP (dimension W T^-1) with government debt (dimension W) only makes sense if we are interested in extracting the T component. The ratio Debt/GDP has dimension T. Whichever way we choose to denominate GDP (per year, per second, per centure), Debt/GDP will be a measure of time. The 125% should really be stated as 1.25 years, or using the per century denomination, 0.0125 centuries. Same thing.

It’s not clear what the proper quantity, 1.25 years, tells us about the economic situation. It’s not like we are going to make everybody work for 1.25 years with zero take-home pay in order to pay off the debt. It’s not even clear whether the debt needs to be paid down.

Maybe it would be better to think of debt in terms of how much it costs to carry it. For instance, at an interest rate of 5%, it costs $$30 $\times 0.05 = \$1.5$ trillion each year to “pay for” the debt. If this money could be invested in the economy or in human capital, it would be a big deal. But we also benefit in an ongoing way from the historical investment of $1.5 trillion/year that was enabled by the debt.

Exercise 13. 14 As of 2025, New York City has an “ambitious” plan to add 50,000 new housing units each year. There are currently about 3,750,000 units in the city. The example about new housing in the text suggests that existing stock falls by about 1% per year and that the number of households in the US is increasing by about 1% per year. Use these numbers to calculate whether 50,000 new units per year will have a significant affect on the housing situation.

Exercise 13. 15 Suppose that the units of a quantity are in “newtons.” (You don’t need to know what a newton is to do this problem, but if you’re interested, a newton is a unit of force.) A relative probability, on the other hand, can be a pure number: dimensionless and with no unit. Accumulate a relative probability distribution on that quantity from $-\infty$ to $\infty$. What will be the units of the accumulated quantity.

To turn a relative probability into an absolute probability, divide the relative probability by the accumulated quantity from the previous paragraph. What will be the units of the absolute probability relevant to our newton-quantity.

An exercise for net present value.

Footnotes

The article mistakenly says “four-year period.”↩︎