Understanding the Cumulative Distribution Function for Age-at-Death

How do we plan for a future that is, by its very nature, uncertain? We buy insurance, we save for retirement, and we build pension plans. But for any of these financial tools to work, someone has to move beyond guessing and start calculating. That ‘someone’ is an actuary, and their primary tool for understanding the biggest uncertainty of all-human lifespan-is mathematics. It might sound a little grim to talk about “age-at-death,” but quantifying this uncertainty is what allows us to build systems that provide financial security for people and their families. At the heart of these calculations are three powerful concepts: the Cumulative Distribution Function (CDF), the Survival Function, and the Probability Density Function (PDF). Let’s break down what they are and how they work together.

What is the Cumulative Distribution Function (CDF)? 🔗

Imagine you have a special rain gauge that measures the total rainfall for an entire year. At the start of the year (January 1st), the gauge reads zero. As the days pass, any rain that falls accumulates. By July 1st, it might show 400mm. By December 31st, it shows the total for the year, say, 1000mm. The reading in this gauge never goes down; it only ever increases or stays the same. This is exactly how the Cumulative Distribution Function, or F(x), works for the age-at-death.

In actuarial science, we use a random variable, let’s call it T, to represent a person’s exact age at death. The CDF, denoted F(x), answers a very specific question: “What is the probability that a newborn individual will die at or before a specific age x?”

Mathematically, we write this as: F(x) = P(T ≤ x)

Just like our rain gauge, the CDF has some logical properties.

• F(0) = 0. The probability of a newborn dying at or before age 0 is zero. (We are considering the moment of birth as the starting point).
• F(x) is non-decreasing. The probability of dying by age 60, F(60), must be greater than or equal to the probability of dying by age 50, F(50). You can’t “undie,” so the cumulative probability can never decrease.
• F(ω) = 1. If we set an ultimate age, (omega), beyond which no one can live (say, 125), the probability of dying at or before this age is 1 (or 100%). Everyone is certain to have died by this limiting age.

So, if a mortality table tells us that F(40) = 0.05, it means there is a 5% chance that a newborn will die sometime between birth and their 40th birthday. If F(70) = 0.22, it means there is a 22% chance a newborn will die at or before age 70. The CDF is our “accumulator” of mortality probability over a lifetime.

Meet the survivor: The Survival Distribution Function 🔗

Now, if F(x) tells us the probability of not making it to a certain age, what about the probability that you do make it? This is where the logical and equally important counterpart, the Survival Distribution Function, or S(x), comes in. As its name suggests, S(x) measures the probability that a newborn survives beyond age x.

It’s the perfect mirror image of the CDF. An event can only end in two ways: a person either dies at or before age x (an event measured by F(x)) or they survive past age x (an event measured by S(x)). Since these are the only two possibilities, their probabilities must add up to 1 (or 100%).

This gives us the most important relationship in survival modeling: S(x) = 1 – F(x).

Let’s use our previous numbers:

• If F(40) = 0.05 (5% chance of dying by 40), then S(40) = 1 – 0.05 = 0.95. This means a newborn has a 95% chance of surviving past their 40th birthday.
• If F(70) = 0.22 (22% chance of dying by 70), then S(70) = 1 – 0.22 = 0.78. A newborn has a 78% chance of living to celebrate more birthdays after 70.

While the CDF is a story of accumulating risk, the survival function is a story of enduring. This function is the cornerstone of the life insurance and pension industries.

• For a life insurer (selling a policy that pays out on death), S(x) is crucial. A high S(x) (high probability of survival) means the company can continue collecting premiums and won’t have to pay the death benefit for a long time.
• For an annuity provider (selling a product that pays you an income for as long as you live), a high S(x) is their biggest risk! It means they will have to keep making payments for many, many years.

Just like F(x), S(x) has logical properties. It’s a non-increasing function. It starts at S(0) = 1 (a newborn has a 100% chance of surviving past age 0) and gradually decreases, eventually hitting S(ω) = 0 (the probability of surviving past the maximum possible age is zero).

[Image: A simple graph showing two curves on the same axes. One curve, labeled 'F(x) - CDF', starts at 0 and rises to 1. The other curve, labeled 'S(x) - Survival Function', starts at 1 and falls to 0. They cross at the 0.5 probability mark, labeled 'Median Age-at-Death'.]

Zooming in: The Probability Density Function (PDF) 🔗

We’ve talked about the total probability accumulated up to age x (the CDF) and the total probability of surviving past age x (the Survival Function). But what about the risk of death at a specific age? What is the “force” of mortality right at the moment a person turns 60, not in the 60 years leading up to it?

This is where the Probability Density Function, or f(x), comes in. If the CDF, F(x), is our “rain gauge” showing the total accumulation, the PDF, f(x), is the “live weather radar” showing the intensity of the rain at any exact moment. It tells us the relative likelihood of death at a precise age x.

In calculus terms, the PDF is the derivative of the CDF. f(x) = F'(x). It measures the rate of change of the cumulative probability.

• Where f(x) is high (e.g., in very old age), it means the CDF is rising very steeply. This implies a high “density” of deaths is occurring at that age.
• Where f(x) is low (e.g., in childhood, after infancy), it means the CDF is rising very slowly, indicating a low density of deaths.

It’s very important to understand that the PDF is not a probability. A common point of confusion, f(x) can have a value greater than 1, while a probability can never be more than 1. It’s a measure of “density,” not a direct probability. Think of it this way: the probability of a person dying at the exact, infinitesimal, infinitely small instant they turn 60 (60.000000…) is zero. There are infinite such instants in a lifetime.

However, the PDF is incredibly useful because we can use it to find the probability of dying within a range of ages. The probability of dying in any specific interval (say, between age 60 and 61) is the area under the PDF curve in that interval. This area, found by using an integral in calculus, is exactly the same as the change in the CDF between those two ages: P(60 < T ≤ 61) = F(61) – F(60).

So, the PDF is the function that, when we “add up” all its values (by integrating), *builds* the CDF. It shows us *where* the mortality risk is concentrated across a lifespan.

The force of mortality 🔗

The PDF is also used to derive a famous actuarial concept: the force of mortality, or hazard rate, often denoted μ(x). This function measures the instantaneous rate of death at age x, given that the individual has already survived to age x. It’s a conditional concept. The formula is μ(x) = f(x) / S(x). In simple terms, it’s the “density” of death at age x (f(x)) divided by the proportion of people still alive at age x to experience that risk (S(x)). This “hazard rate” is what actuaries often model directly to build their mortality tables.

Putting it all into practice 🔗

Let’s see how these three functions, F(x), S(x), and f(x), work together to answer real-world questions.

Example 1: The life insurance question 🔗

A life insurance company is selling a “term life” policy to a 20-year-old. The policy term is 40 years, meaning it pays out if the person dies before age 60. The company needs to know the probability of this event.

We are not asking for F(60), which is the probability a newborn dies by 60. We are asking for the probability that a person who is already 20 dies before 60. This is a conditional probability.

Here’s how we think it through:

1. We are only interested in the group of people who have already survived to age 20. The probability of this is S(20). This is our new “100%” or our starting population.
2. We want to find the probability of dying *between* age 20 and age 60. Using the CDF, this is the total probability of dying by 60, F(60), minus the probability of having *already* died by 20, F(20). So, the probability is F(60) – F(20).
3. To get the conditional probability, we divide the event we’re interested in (dying between 20 and 60) by the population we’re starting with (those who survived to 20).

The probability is: (F(60) – F(20)) / S(20)

Since S(x) = 1 – F(x), we can also write S(20) as 1 – F(20). This single formula, derived from the CDF and Survival Function, is the basis for pricing a multi-million dollar insurance industry.

Example 2: Is this a valid CDF? 🔗

An analyst proposes a new mathematical model for F(x). How do we know if it’s a “valid” CDF for a lifetime distribution? We just check the basic rules we established earlier.

Let’s say the proposed function is F(x) = (x/100)² for ages 0 to 100, and F(x) = 1 for ages over 100.

1. Does F(0) = 0? Yes. F(0) = (0/100)² = 0. That’s a good start.
2. Is F(x) non-decreasing? Yes. As x gets bigger, (x/100)² also gets bigger. For example, F(20) = 0.04 and F(30) = 0.09. The probability is always increasing, which makes sense.
3. Does F(ω) = 1? Yes. At the limiting age (ω=100), F(100) = (100/100)² = 1. The model correctly states that the probability of dying by age 100 is 100%.

Since the function meets these three criteria, it is a valid (though very simple) Cumulative Distribution Function. From this, we could immediately find the other functions.

• The Survival Function would be S(x) = 1 – (x/100)².
• The PDF, f(x), would be the derivative: f(x) = 2x / 10000.

These three functions-the accumulator (CDF), the mirror-image survivor (S(x)), and the instantaneous rate (PDF)-are the foundational toolkit for modeling and understanding mortality. They allow actuaries and economists to turn the profound uncertainty of life into a manageable risk, enabling the financial systems that support us all.

What do you think?

When you think about long-term financial planning, like for retirement, do you find it more intuitive to think about your probability of surviving (the Survival Function) or the chances of not surviving (the Cumulative Distribution Function)? Why do you think one feels more natural to you than the other?

References

1. https://en.wikipedia.org/wiki/Survival_function
2. https://web.stanford.edu/~lutian/coursepdf/unit1.pdf
3. https://egyankosh.ac.in/bitstream/123456789/88685/1/Unit-10.pdf
4. https://www.soa.org/493459/globalassets/assets/library/research/transactions-of-society-of-actuaries/1961/january/tsa61v13pt1n36ab9.pdf
5. https://faculty.washington.edu/yenchic/18W_425/Lec5_survival.pdf