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De Moivre's law
Definition
De Moivre's law is a simple actuarial survival model in which lifetime is uniformly distributed between zero and a specified limiting age. Its survival curve decreases linearly to zero at that limit. Conditional on surviving to age x, the remaining lifetime is uniform between zero and the limiting age minus x. It is a mathematical model, not a universal law of human longevity.
What it means
The model assumes equal probability density across the permitted ages at death. In a large hypothetical starting cohort, the expected number of deaths per equal age interval is therefore constant, while the expected number of survivors falls linearly.
This does not imply a constant risk among those still alive. The same expected number of deaths is drawn from an increasingly small surviving population, so the conditional mortality rate increases with age.
In this entry, the term refers to the survival model. It is distinct from de Moivre's formula for complex numbers and from the de Moivre - Laplace normal approximation. The model is useful for understanding survival mathematics and simple benchmarks, but realistic applications require evidence about its fit.
How it works
Write the limiting age as omega. For age x between zero and omega, the survival probability from birth is S(x) = 1 − x/omega. The density is 1/omega within that interval, and the force of mortality is 1/(omega−x).
For someone already alive at age x, divide S(x+t) by S(x). The probability of surviving another t years is (omega−x−t)/(omega−x), for 0 ≤ t ≤ omega−x. The probability of death during that interval is t/(omega−x).
Because remaining lifetime is uniform on an interval of length omega−x, its complete expectation is (omega−x)/2. This expectation includes fractional years. It is different from the expected number of completed future years. All these formulas depend on the stated model and require their domain restrictions; extending the linear survival formula beyond omega would create impossible negative probabilities.
Three practical examples
- An actuarial student derives conditional survival probabilities from a simple survival curve and checks the distinction between age at death and remaining lifetime.
- A model-testing team uses a uniform lifetime benchmark to confirm that software handles limiting ages and conditional probabilities correctly.
- A business analytics class compares the model's increasing hazard with an exponential model's constant hazard to understand how lifetime assumptions change forecasts.
Think of it this way
Think of a hypothetical cohort represented by a strip that loses the same length at each step. The strip becomes shorter at a constant absolute rate, but each removed piece is a growing fraction of what remains. This illustrates why a linear survival curve can correspond to an increasing conditional risk rather than a constant one.
Worked example
Consider an original teaching model with omega = 96 years and a person known to be alive at age 40. These invented assumptions are not a life expectancy estimate for any real person or population.
The remaining lifetime is uniform from zero to 56 years. The probability of surviving another 14 years is (96−40−14)/(96−40) = 42/56 = 0.75. The complementary probability of dying within those 14 years is 14/56 = 0.25. Expected remaining lifetime is 56/2 = 28 years.
The unconditional probability of surviving from birth to age 54 is S(54) = 42/96 = 0.4375. This differs from 0.75 because the conditional calculation already knows that survival to age 40 occurred. Dividing 0.4375 by S(40) = 56/96 recovers 0.75.
The force of mortality at age 40 is 1/56, approximately 0.01786 per year. At age 80 it is 1/16 = 0.0625 per year. The density at death remains constant within the model, while this conditional instantaneous rate rises. A rate is not generally an interval probability, even though this particular model's one-year death probability at an integer age numerically equals 1/(omega−x).
Illustrative case study
In this fictional case, Sequoia Analytics develops teaching software for survival models. A test workbook uses de Moivre's law with an invented limiting age to verify formulas for survival, remaining lifetime, and expected duration.
The team notices that an early implementation returns a negative survival probability beyond the limit. It fixes the piecewise definition and adds checks that conditional calculations divide by the probability of reaching the starting age. Its documentation clearly identifies the exercise as a model demonstration.
The case illustrates software verification and assumption transparency, not predictive adequacy.
Common misconceptions and mistakes
Uniform lifetime density does not mean constant hazard. The exponential distribution has constant hazard; the uniform model considered here does not.
A limiting age is a model parameter, not an observed universal maximum that applies to everyone. Do not infer individual prognosis from a classroom example or treat a mathematical endpoint as evidence about biology.
The assumption of uniform distribution of deaths within individual life-table intervals is also different from imposing one uniform lifetime distribution across the entire lifespan. Local interpolation can be used with a much more flexible overall survival pattern.
Complete life expectancy is not a guaranteed remaining duration. Individuals have a distribution of possible remaining lifetimes rather than one predetermined value.
Business use
For businesses, the main value is conceptual clarity and a tractable test case. Analysts can check whether forecasting systems distinguish unconditional survival from survival conditional on an existing relationship or elapsed service time.
Before applying any lifetime model operationally, define the event, population, time origin, and treatment of incomplete observation. Compare the proposed survival and hazard patterns with relevant evidence. If data support changing risk patterns, use a model capable of representing them. Simple formulas are helpful for checking calculations, but computational convenience does not establish that a model describes the business process.
People also ask
Does the model predict constant annual deaths?
It gives constant expected deaths per equal interval in a fixed starting cohort under its assumptions, not identical realized counts.
Is the hazard constant?
No. It rises as age approaches the limiting age.
Is this the complex-number formula?
No. The name here identifies an actuarial mortality model.
Check your understanding
Check your understanding: Why is conditional survival to age 54 equal to 0.75 in the example?
Answer: Survival to age 40 is already known.
What does 28 years represent?
Answer: Mean remaining lifetime under the stipulated model.
Why must the formula stop at the limit?
Answer: Survival probabilities cannot become negative.
Does a constant density imply constant conditional risk?
Answer: No.
Related terms
- Survival function
- Uniform distribution
- Hazard rate
- Life table
- Conditional probability
- Censoring
- force of mortality
- complete life expectancy
- exponential distribution
Learn next
Derive conditional survival from a general survival function. Compare uniform and exponential lifetime models, then study life-table interpolation and evidence-based survival estimation.
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Sources
Sources: Emiliano Valdez, Survival Models, Michigan State University lecture notes, https://users.math.msu.edu/users/valdezea/stt455f14/STT455Weeks2to3-F2014-annot.pdf. ACTEX, Survival Models, section on uniform lifetime distributions and their limitations, https://actexlearning-test.azurewebsites.net/samples/ACTEX-Survival_Models-SAMPLE.pdf. The numerical model and fictional software case are original.
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