Editing Survival analysis (section)

===Quantities derived from the survival distribution===

'''Future lifetime''' at a given time <math>t_0</math> is the time remaining until death, given survival to age <math>t_0</math>. Thus, it is <math>T - t_0</math> in the present notation. The '''expected future lifetime''' is the [[expected value]] of future lifetime. The probability of death at or before age <math>t_0+t</math>, given survival until age <math>t_0</math>, is just

<math display="block">P(T \le t_0 + t \mid 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)}.</math>
Therefore, the probability density of future lifetime is

<math display="block">\frac{d}{dt}\frac{F(t_0 + t) - F(t_0)}{S(t_0)} = \frac{f(t_0 + t)}{S(t_0)}</math>
and the expected future lifetime is

<math display="block">\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t_0+t)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt,</math>
where the second expression is obtained using [[integration by parts]].

For <math>t_0 = 0</math>, 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 '''[[median]] lifetime''', for which ''q'' = 1/2, or other quantiles such as ''q'' = 0.90 or ''q'' = 0.99.