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:

Since

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:

Tue Jun 6 00:15:46 DF 1995