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For the force of mortality, why do we denote it as mu (x+t) [for example] and not just mu (x)?

+1 vote
asked Mar 17 in BUS 3018F - Models by anonymous

Isn't it just supposed to represent the instantaneous rate of death at age x? So why do we need to add the t?

1 Answer

+4 votes
answered Mar 18 by asilmotala (2,380 points)
selected Mar 20 by Rowan
Best answer

So firstly we need to consider what mu represents, which is force of mortality at a specific age of time. Furthermore we need to consider surrounding information, which in the case of mortality models is the age of the individual (x) and the time after the age which we are interested in (t).

Thus the force of mortality becomes a function of both age (x) and time (t) and is represented by mu (x+t) and not just mu (x). To simplify this, I will put it in the context of an example, so you can see it's relevance.

Example We have a 40 year old male, and we are interested his probability of survival till age 60. Thus we set x at 40, and and t varies from 0 to 20. The answer would therefore be as follows:

$$e^{\int_{0}^{20} \mu(40+t) \space dt} $$

Alternatively we could have:

$$e^{\int_{40}^{60} \mu(x) \space dx} $$

I do hope his clarifies your issue.

commented Mar 18 by michwairish (320 points)

Okay, I understand that. In question 1.1 of the tut pack, it reads as follows:

  1. 1.1  Suppose you know only that, for a certain life, μ40 = 0.00922 and μ41 = 0.00966. Use these figures to find a simple approximation to q40.
    (Hint: how is
    p40 related to μ40+t, 0 t < 1?)

  2. Do we need μ41 in this case? Or do we just use μ40 to calculate p40 and then put q40=1-p40 ? Or is μ40+s a linear function between μ40 and μ41 ?

commented Mar 18 by asilmotala (2,380 points)

So the key to this question is that they ask you for an approximation of q40. This it means you need to make some sort of assumption about mu (40+t) for 0 <t <1. The key part to this question is to state your assumption at the beginning, as this shows the marker that you are making an approximation since the exact data is unknown, and it also shows which assumption is made when answering the question. You could assume a constant force of mortality and use mu (40) alone to calculate p40, and then calculate q40. However it would be better to assume linearity as you have suggested, and calculate p40 under the assumption of linearity (See Trapezoidal Rule for Integration Approximation), and from that calculate q40.

commented Mar 18 by michwairish (320 points)

Okay, thank you!