Quantities derived from the survival distribution

Manoj Monday 21 May 2012

Future lifetime at a given time $t_0$ is the time remaining until death, given survival to age $t_0$ . Thus, it is $T - t_0$ in the present notation. The expected future lifetime is the expected value of future lifetime. The probability of death at or before age $t + t_0$ , given survival until age $t_0$ , is just

$P(T \le t_0 + t | T > t_0) = \frac{P(t_0 < T \le t_0 + t)}{P(T > t_0)} = \frac{F(t_0 + t) - F(t_0)}{S(t_0)}.$

Therefore the probability density of future lifetime is

$\frac{d}{dt}\frac{F(t_0 + t) - F(t_0)}{S(t_0)} = \frac{f(t_0 + t)}{S(t_0)}$

and the expected future lifetime is

$\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t+t_0)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt,$

where the second expression is obtained using integration by parts.

For $t_0 = 0$ , that is, at birth, this reduces to the expected lifetime.

In reliability problems, the expected lifetime is called the mean time to failure, and the expected future lifetime is called the mean residual lifetime.

As the probability of an individual surviving until age t or later is S(t), by definition, the expected number of survivors at age t out of an initial population of n newborns is n × S(t), assuming the same survival function for all individuals. Thus the expected proportion of survivors is S(t). If the survival of different individuals is independent, the number of survivors at age t has a binomial distribution with parameters n and S(t), and the variance of the proportion of survivors is S(t) × (1-S(t))/n.

The age at which a specified proportion of survivors remain can be found by solving the equation S(t) = q for t, where q is the quantile in question. Typically one is interested in the medianlifetime, for which q = 1/2, or other quantiles such as q = 0.90 or q = 0.99.

One can also make more complex inferences from the survival distribution. In mechanical reliability problems, one can bring cost (or, more generally, utility) into consideration, and thus solve problems concerning repair or replacement. This leads to the study of renewal theory and reliability theory of aging and longevity.

Thanks for reading Quantities derived from the survival distribution

1 comment:

MS22 May 2012 at 19:53
this notes useful to me! thank you!
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