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Life expectancy

The mean time spent in each state between time [0,n] is an important synthetic index. It depends of the initial age x and initial status j.

Let , be a random variable describing the status of an individual at time t, if initially at age x and in status j at time 0. We have:

If we now introduce

, is the proportion of time n spent in status k. This again is a random variable, for which expectancy, or ``life expectancy'' can be written:


Generally life expectancy is not enough to synthesize the whole distribution, particularly when variance is high. The variance can be computed as:


we find

For symetrical reasons, one can integrate on a subdiagonal , and this leads with , to:

Since the process is markovian

The variance can thus finally be written:

Other important relations, such as the disaggregation into life expectancy with disability and disability-free life expectancy can be deduced:

Nicolas Brouard
Tue Jun 6 00:15:46 DF 1995