Concepts of survival models: formula sheet

Key formulas

Distribution function of T_x

F_x(t) = P(T_x ≤ t) = tq_x

Probability of death within t years for a life aged x.

Survival function of T_x

S_x(t) = P(T_x > t) = tp_x = 1 − F_x(t)

S_x(0) = 1 and S_x is non-increasing.

Density of T_x

f_x(t) = F_x'(t) = −S_x'(t) = tp_x · μ(x+t)

Valid where the derivative exists, for t ≥ 0.

Interval probability

P(s < T_x ≤ t) = F_x(t) − F_x(s) = sp_x − tp_x

For 0 ≤ s < t. Equivalently the integral of f_x from s to t.

Deferred death probability

s|tq_x = P(s < T_x ≤ s+t) = sp_x − (s+t)p_x = sp_x · tq_(x+s)

Survive s years, then die within the next t.

Link to survival from birth

tp_x = S_0(x+t) ÷ S_0(x)

Conditional on being alive at age x.

Curtate future lifetime

K_x = ⌊T_x⌋; P(K_x = k) = kp_x · q_(x+k) = kp_x − (k+1)p_x

k = 0, 1, 2, …

Survival of K_x

P(K_x ≥ k) = P(T_x ≥ k) = kp_x

Also P(K_x > k) = (k+1)p_x.

Definition of force of mortality

μx = lim(h→0+) (h q_x) ÷ h = f_x(0) ÷ S_x(0)

Here f_x and S_x refer to the future lifetime T_x. It is a rate, not a probability.

In terms of the survival function of age at death

μx = −S′(x) ÷ S(x) = −d/dx ln S(x)

S(x) = P(X > x) for lifetime X from birth, or l_x up to a constant: μx = −(1/l_x) · dl_x/dx.

Survival probability from force of mortality

tpx = exp(−∫ from 0 to t of μ(x+s) ds)

Holds for any t ≥ 0 where μ is integrable. For constant μ, tpx = e^(−μt).

Density of future lifetime

f_x(t) = tpx · μ(x+t)

Density of T_x. Also equals −d/dt (tpx).

Deferred death probability

t|u qx = tpx − (t+u)px = ∫ from t to t+u of spx · μ(x+s) ds

Probability of surviving t years then dying in the next u years.

Cumulative hazard

Λ_x(t) = ∫ from 0 to t of μ(x+s) ds, so tpx = e^(−Λ_x(t))

Useful when the integral is simple.

Survival probability

tpx = lx+t ÷ lx

Probability that a life aged x survives to age x+t. Denominator is always the starting age.

Death probability

tqx = 1 − tpx = (lx − lx+t) ÷ lx

Probability of dying within t years of age x.

Annual deaths

dx = lx − lx+1 = lx × qx

Expected deaths between exact ages x and x+1.

One-year probabilities

px = lx+1 ÷ lx and qx = dx ÷ lx

These are tpx and tqx with t = 1.

Deferred mortality

t|uqx = tpx × uqx+t = (lx+t − lx+t+u) ÷ lx

Survive t years, then die within the next u years.

Deferred mortality as a difference

t|uqx = t+uqx − tqx = tpx − t+upx

Useful when you are given survival or death probabilities rather than l values.

Multiplication rule for survival

t+upx = tpx × upx+t

Survive to x+t first, then survive u more years from age x+t.

Sum of deaths

lx − lx+n = dx + dx+1 + … + dx+n−1

Use when you are given dx values instead of lx values.

Complete expectation of life

e°x = E[Tx] = ∫0^∞ t px dt

Equivalent to ∫0^∞ t · t px · μ(x+t) dt. The survival form is usually quicker.

Curtate expectation of life

ex = E[Kx] = Σ(k=1 to ∞) k px

Sum starts at k = 1, not 0.

Curtate probability mass function

P(Kx = k) = k px × q(x+k) = k| qx

Probability of surviving k years then dying in the next year.

Life table form

ex = Σ(k≥1) l(x+k) ÷ lx

Use the table directly. Stop where l becomes zero.

Relationship under uniform distribution of deaths

e°x ≈ ex + ½

Exact if deaths are uniform within each year of age. Otherwise only an approximation.

Recursion for curtate expectation

ex = px × (1 + e(x+1))

Useful for working backwards from the oldest age.

Recursion for complete expectation

e°x = ∫0^1 t px dt + px × e°(x+1)

The first term is the expected time lived in the first year.

Temporary complete expectation

e°x:n| = ∫0^n t px dt

Expected years lived within the next n years.

Gompertz law

μx = B c^x and tpx = exp[ −B c^x (c^t − 1) ÷ ln c ]

Parameters B > 0 and c > 1. Obtained by integrating μ(x+u) = B c^(x+u) from u = 0 to t.

Makeham law

μx = A + B c^x and tpx = exp[ −A t − B c^x (c^t − 1) ÷ ln c ]

Gompertz plus a constant A. Equivalent form: tpx = s^t g^(c^x (c^t − 1)), where s = exp(−A) and g = exp(−B ÷ ln c).

Makeham form of lx

lx = k s^x g^(c^x)

Useful for ratios like l(x+t) ÷ lx. The constant k cancels in any survival probability.

UDD: probability of death

s qx = s × qx for 0 ≤ s ≤ 1

So s px = 1 − s qx. Also l(x+s) = (1 − s) lx + s l(x+1), a straight line between integer ages.

UDD: force of mortality

μ(x+s) = qx ÷ (1 − s qx) for 0 ≤ s < 1

Force rises through the year. The density s px × μ(x+s) = qx is constant.

UDD: survival from a fractional age

t q(x+s) = t qx ÷ (1 − s qx) for s + t ≤ 1

If s + t > 1, split at the next integer age and multiply the pieces.

Constant force of mortality

s px = (px)^s = e^(−μ s), with μ = −ln(px) for 0 ≤ s ≤ 1

Force is the same throughout the year. Also t p(x+s) = (px)^t for s + t ≤ 1, which does not depend on s.

Expectation of life under UDD

e°x = ex + ½

Complete expectation equals curtate expectation plus one half, assuming UDD in every year of age.

Select-period notation

q[x]+t = P(a life aged x + t, selected at age x, dies within 1 year)

Valid for t = 0, 1, 2, ... Here t is the duration since selection.

Ultimate rule

q[x]+t = q(x+t) for t ≥ r

r is the select period. The ultimate table depends on attained age only.

Select survival probabilities

p[x]+t = 1 − q[x]+t; l[x]+t+1 = l[x]+t × p[x]+t

Use for any t, but move to the ultimate column when t reaches r.

Multi-year survival from a select age

tp[x] = l[x]+t ÷ l[x] = p[x] × p[x]+1 × ... × p[x]+t−1

After r years, use ultimate lives: for t > r, tp[x] = (l(x+t) ÷ l(x+r)) × (l[x]+r ÷ l[x]), and l[x]+r = l(x+r) only in a consistent table.

Consistency of a select table

l[x]+r = l(x+r)

Holds when the table's ultimate l column is defined to begin at the end of the select period.

Quick revision

• T_x is the future lifetime of a life aged x; S_x(t) = P(T_x > t) = tp_x.
• μ_x = f_x(0) ÷ S_x(0), and in general μ_{x+t} = f_x(t) ÷ S_x(t).
• tp_x = exp(−∫₀ᵗ μ_{x+s} ds).
• f_x(t) = tp_x × μ_{x+t}.
• From a life table: tp_x for whole t = l_{x+t} ÷ l_x, and q_x = d_x ÷ l_x with d_x = l_x − l_{x+1}.
• Deferred probability: t|uq_x = tp_x − t+up_x.
• Complete expectation e°_x = ∫₀^∞ tp_x dt; curtate e_x = Σ (k from 1) kp_x.
• Under uniform distribution of deaths, tq_x = t × q_x for 0 ≤ t ≤ 1, and μ_{x+t} = q_x ÷ (1 − t q_x).
• Under constant force of mortality over the year, tp_x = (p_x)^t.
• Gompertz: μ_x = Bc^x. Makeham: μ_x = A + Bc^x.
• Select mortality: use q_[x]+r for a life selected at age x, with duration r; after the select period it matches ultimate rates.
• State your assumptions clearly in written answers, especially for fractional ages.

Common mistakes

• Treating P(K_x = k) as kp_x. Fix: kp_x is P(K_x ≥ k). For exactly k, subtract: P(K_x = k) = kp_x − (k+1)p_x.
• Confusing P(K_x > k) with P(K_x ≥ k). Fix: Convert to T_x. K_x ≥ k means T_x ≥ k, so probability kp_x. K_x > k means K_x ≥ k+1, so probability (k+1)p_x.
• Treating μx as a probability and saying it cannot exceed 1. Fix: Remember that μx is a rate per year. It is non-negative but can be above 1. qx and tpx are probabilities between 0 and 1, whereas μx and densities are rates or densities and are not bounded by 1.
• Writing qx = μx. Fix: qx = 1 − exp(−∫ from 0 to 1 of μ(x+s) ds). Even for constant μ, qx = 1 − e^(−μ), which is only approximately μ when μ is small.
• Dividing by lx+t instead of lx when computing t|uqx. Fix: Ask which age the life is known to be alive at. For t|uqx it is age x, so divide by lx. Only use lx+t if the question says 'given alive at x+t'.
• Reading t|uqx as the probability of dying between ages t and u. Fix: Read it as: defer t years, then a window of u years. The window runs from x+t to x+t+u.
• Starting the curtate sum at k = 0. Fix: Remember ex = Σ from k = 1. Kx ≥ k has probability k px, and k = 0 adds nothing.
• Stating e°x = ex + ½ as an exact result. Fix: Write 'assuming deaths uniformly distributed over each year of age' and use ≈, unless the law is exactly uniform in each year.
• Using t q(x+s) = t qx under UDD when the start age is not an integer. Fix: Use t q(x+s) = t qx ÷ (1 − s qx). The denominator is the probability of surviving to x+s.
• Believing the force of mortality is constant under UDD. Fix: Under UDD the density s px × μ(x+s) = qx is constant, but μ(x+s) = qx ÷ (1 − s qx) increases with s. Only the constant force assumption has constant μ.

Exam tips

• Always convert K_x events to T_x events before writing any probability. It removes most inequality errors.
• Show the link S_x(t) = S_0(x+t) ÷ S_0(x) explicitly. Examiners give method marks for the conditional step.
• Keep full precision in exponentials until the final line, then round to the accuracy asked.
• In multiple-choice questions, test candidate answers with S_x(0) = 1 and non-negativity of f_x to eliminate wrong options quickly.
• Use the notation tp_x, tq_x and μ(x+t) in written answers, and define any symbol you introduce.
• Show the integral and the exponential before substituting numbers. Method marks are given for the setup.
• If a question says 'derive', start from the definition with h q_x and limits. Do not jump to the result.
• For piecewise μ, draw a quick timeline of the age bands. It prevents missed intervals.