ACTL10001 Chap.6 Mortality and Life Tables
Mortality and Life Tables
Mortality models describe lifetime through survival probabilities, death probabilities and the force of mortality. A survival function gives the probability of reaching each age from birth. For someone already aged x, future survival must be conditioned by dividing survival to the later age by survival to x. Life tables express the same experience through survivor counts, deaths and one year probabilities.
The chapter moves between continuous survival models and discrete table identities, with particular attention to conditioning, interval endpoints and model suitability. Direction checks and probability bounds help expose a missing denominator or reversed subtraction.
What this chapter covers
- 01
Survival functions
- 02
Lifetime distributions
- 03
Conditional survival
- 04
Deferred death probabilities
- 05
Force of mortality
- 06
Life table survivor counts
- 07
Death and survival probabilities
- 08
Mortality model critique
Worked example · free
Conditional survival from a life table
- 1Ten year survival from age 40 is l50/l40.
- 184,600/90,000 = 0.9400.
- 1Ten year survival from age 50 is l60/l50.
- 173,602/84,600 = 0.8700.
Key terms
- Survival Function
- The probability that lifetime from birth exceeds a stated age.
- Conditional Survival
- Future survival probability given that the life has already reached the starting age.
- Death Probability
- The chance of death within a specified future interval for a life at a stated age.
- Mortality Force
- The instantaneous mortality intensity among lives who have survived to the stated age.
- Life Table
- A table connecting survivors, deaths and survival or death probabilities by age.
- Limiting Age
- The age boundary beyond which the life table assumes no survivors remain.
Mortality and Life Tables FAQ
Why do conditional probabilities use a ratio?
The life is already known to have survived to the starting age, so later survival is measured relative to that surviving group rather than the birth cohort.
How do survivor counts give death probabilities?
Deaths between ages are the fall in survivor counts. Divide that fall by survivors at the opening age to obtain the one year death probability.
What does a constant force imply?
Equal length future intervals have the same conditional survival regardless of attained age, which is convenient mathematically but unrealistic across a full human lifespan with age-varying mortality.
What checks should I run?
Probabilities must lie between zero and one, survival plus death over the same period must equal one and survivor counts cannot increase in a closed life table.
What is the fastest way to catch a reversed survival ratio?
Check that conditional survival is between zero and one and decreases as the horizon lengthens. The numerator represents survival to the later age and cannot exceed the denominator representing survival to the attained age.
How can I validate a completed life-table row?
Check that deaths plus survivors at the next exact age equal survivors at the current age, and that the one-year survival and death probabilities sum to one. For a multi-year interval, compare the survivor ratio with the product of consecutive one-year survival probabilities. These independent identities expose rounding drift and displaced age labels.
Retain unrounded expected survivor counts internally when the table is not restricted to literal whole lives. Label every probability with its starting age and horizon.
Exam move
Draw age intervals and mark the conditioning age before substituting any survival function. Practise translating among s(x), conditional survival, death within a period and deferred death. Build short life tables forward from q values and backward from l values, checking every row with l equals deaths plus next survivors. Compare simple mortality laws by describing the age implication they impose.
State whether a model criticism applies globally or only outside a limited fitted range. Anchor every probability on an age line. Mark the attained age, the end of the survival interval and any later death interval before writing notation. Translate the symbols into words so that conditional survival is not confused with newborn survival to the same terminal age.
Verify nested survival using the chain rule and confirm that longer-horizon survival cannot exceed shorter-horizon survival. When differentiating a survival function, keep the negative sign in the force of mortality and check that the resulting intensity is nonnegative. Build life-table rows in a fixed order from survivors to deaths, death probability and next survivors.
Reconcile each row locally, and verify multi-year survival both from survivor ratios and products of annual probabilities. Retain expected survivor counts without premature rounding when the table is theoretical. Create short exercises that move in both directions: derive q from adjacent survivor counts, then reconstruct the missing survivor count from q.
For mortality laws, calculate a few ages and describe the implied age pattern instead of accepting a formula solely because it produces valid probabilities. Distinguish a mathematical property such as constant force from an empirical claim about real mortality. End every model answer with its domain and limitation.
A simplified exponential model can illustrate conditional probability cleanly while still failing to represent mortality that rises with attained age.
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