By D C M Dickson; Mary Hardy; H R Waters

Balancing rigour and instinct, and emphasizing functions, this contemporary textual content is perfect for collage classes and actuarial examination preparation.

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**Sample text**

Calculate t p30 for t = 1, 5, 10, 20, 50, 90. Calculate t q40 for t = 1, 10, 20. Calculate t |10 q30 for t = 1, 10, 20. Calculate ex for x = 70, 71, 72, 73, 74, 75. ◦ Calculate ex for x = 70, 71, 72, 73, 74, 75, using numerical integration. 5 Let F0 (t) = 1 − e−λt , where λ > 0. (a) (b) (c) (d) Show that Sx (t) = e−λt . Show that µx = λ. Show that ex = (eλ − 1)−1 . What conclusions do you draw about using this lifetime distribution to model human mortality? 02, calculate (a) (b) (c) (d) (e) px+3 , , p 2 x+1 , 3 px , 1 |2 qx .

This is the curtate future lifetime. We can ﬁnd the probability function of Kx by noting that for k = 0, 1, 2, . , Kx = k if and only if (x) dies between the ages of x + k and x + k + 1. Thus for k = 0, 1, 2, . . Pr[Kx = k] = Pr[k ≤ Tx < k + 1] = k |qx = k px − k+1 px = k px − k px px+k = k px qx+k . The expected value of Kx is denoted by ex , so that ex = E[Kx ], and is referred to as the curtate expectation of life (even though it represents the expected curtate lifetime). So E[Kx ] = ex ∞ = k Pr[Kx = k] k=0 ∞ = k (k px − k+1 px ) k=0 = (1 px − 2 px ) + 2(2 px − 3 px ) + 3(3 px − 4 px ) + · · · ∞ = k px .

1988). Both Gompertz’ law and Makeham’s law are special cases of the GM formula. 3, we noted the importance of the force of mortality. A further signiﬁcant point is that when mortality data are analysed, the force of mortality 36 Survival models is a natural quantity to estimate, whereas the lifetime distribution is not. The analysis of mortality data is a huge topic and is beyond the scope of this book. An excellent summary article on this topic is Macdonald (1996). For more general distributions, the quantity f0 (x)/S0 (x), which actuaries call the force of mortality at age x, is known as the hazard rate in survival analysis and the failure rate in reliability theory.