Future Lifetime Distributions for Two Lives: First and Last Death

Understand Future Lifetime Distributions for Two Lives

Key rules to remember

Time to first and last death

T_xy = min(T_x, T_y); T_x̄ȳ = max(T_x, T_y)

Also T_xy + T_x̄ȳ = T_x + T_y. The curtate versions are K_xy = min(K_x, K_y) and K_x̄ȳ = max(K_x, K_y).

Joint life survival probability

t p_xy = t p_x × t p_y

Needs independence. Then t q_xy = 1 − t p_x × t p_y.

Last survivor survival probability

t p_x̄ȳ = t p_x + t p_y − t p_xy

Holds for any joint distribution of (T_x, T_y). Under independence, t q_x̄ȳ = t q_x × t q_y.

Force of mortality of the joint status

μ_xy(t) = μ_x+t + μ_y+t

Under independence. Density of T_xy: f(t) = t p_xy × (μ_x+t + μ_y+t).

Density of the time to last death

f(t) = t p_x μ_x+t + t p_y μ_y+t − t p_xy (μ_x+t + μ_y+t)

Differentiate −t p_x̄ȳ with respect to t. Under independence, this equals the derivative of t q_x × t q_y.

Complete expectation of life

e°_xy = ∫₀^∞ t p_xy dt; e°_x̄ȳ = e°_x + e°_y − e°_xy

e°_x̄ȳ uses t p_x̄ȳ in the integral. The relation follows because t p_x̄ȳ = t p_x + t p_y − t p_xy.

Curtate expectation of life

e_xy = Σ (k = 1 to ∞) k p_xy; e_x̄ȳ = e_x + e_y − e_xy

e_x̄ȳ uses k p_x̄ȳ in the sum. Always start the sum at k = 1.

Probability of death in a given year

P(K_xy = k) = k p_xy − (k+1) p_xy

The same form works for the last survivor with k p_x̄ȳ. K is the curtate future lifetime.

Approximation linking curtate and complete

e°_xy ≈ e_xy + ½

Valid only under the assumption of uniform distribution of deaths within each year. It is an approximation, not an identity.

How to solve Future Lifetime Distributions for Two Lives questions

1. 1Identify the status. Joint life (xy) means first death and the minimum. Last survivor (x̄ȳ) means second death and the maximum.
2. 2State your assumption clearly. Write 'assume T_x and T_y are independent' unless the question gives a dependence.
3. 3Write the survival function. Use t p_xy = t p_x × t p_y for the joint life. For the last survivor, use t p_x̄ȳ = t p_x + t p_y − t p_xy, or t q_x̄ȳ = t q_x × t q_y.
4. 4If a density is needed, differentiate the distribution function. For T_xy, use f(t) = t p_xy (μ_x+t + μ_y+t).
5. 5For an expectation, integrate t p over t (complete) or sum k p from k = 1 (curtate). Use the status asked about.
6. 6For a last survivor expectation, find e_x, e_y and e_xy separately, then use e_x̄ȳ = e_x + e_y − e_xy.
7. 7Check the units and bounds. Is the answer in years? Is e_xy ≤ min(e_x, e_y)? Is e_x̄ȳ ≥ max(e_x, e_y)?

Quickest way: Build the joint table first, then use e_x̄ȳ = e_x + e_y − e_xy

Common mistakes in Future Lifetime Distributions for Two Lives

• Using t p_xy = t p_x + t p_y or taking the average of the two lifetimes.

  Students mix up 'at least one alive' with 'both alive'.

  Fix: Joint life means both alive, so multiply (under independence). The addition-and-subtract form belongs to the last survivor only.

• Writing t q_x̄ȳ = t q_x + t q_y.

  Students copy the pattern of the joint life formula without checking which event it describes.

  Fix: A last survivor status has failed only when both lives are dead, so multiply the death probabilities: t q_x̄ȳ = t q_x × t q_y (independence).

• Starting the curtate sum at k = 0, or integrating from the wrong limits.

  Students remember that t p = 1 at t = 0 and include it.

  Fix: Curtate expectation is Σ from k = 1. Complete expectation integrates from 0 to ∞ (or to the limiting age).

• Adding μ_x and μ_y for the last survivor status.

  The joint life rule μ_x+t + μ_y+t is applied to every status.

  Fix: The sum of forces applies to the joint life status only. For the last survivor, work from the survival probability and differentiate.

• Applying e_x̄ȳ = e_x + e_y − e_xy but mixing curtate and complete values.

  Tables give e_x curtate, and a formula uses e°_xy.

  Fix: Use one type throughout. If you must convert, state the approximation e° ≈ e + ½ and apply it consistently.

• Forgetting to state the independence assumption.

  Students treat it as obvious.

  Fix: Write it as the first line of your solution. Marks are usually awarded for stating the assumption.

Worked examples