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Let $S_{0}(t) = (1-\frac{t}{105})^{1/5}$ be the survival function of a newborn. What is the median future lifetime at age $50$? So $S_{50}(t) = \frac{S_{0}(50+t)}{S_{0}(50)} = \frac{\left(1-\frac{50+t}{105}\right)^{1/5}}{ \left(1-\frac{50}{105}\right)^{1/5}}$

The median future lifetime would be the value of $s$ such that $ \int_{0}^{s} \frac{\left(1-\frac{50+t}{105}\right)^{1/5}}{ \left(1-\frac{50}{105}\right)^{1/5}} \ dt = 0.5$

Is that correct?

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    @joriki Silly me. I have removed the incorrect comment.2012-07-20

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What you did would be correct if this were the probability density function for the time of death. However, the survival function is a function specifying the probability that the newborn will still be alive at time $t$, which is the complement of the cumulative distribution function of the time of death. Then the median future lifetime at age $50$ is just the lifetime beyond $50$ at which half the people who were still alive at $50$ have died, that is,

$ \begin{align} \left(1-\frac{50+t}{105}\right)^{1/5} &= \frac12\left(1-\frac{50}{105}\right)^{1/5}\;, \\ 1-\frac{50+t}{105} &= \frac1{32}\left(1-\frac{50}{105}\right) \;,\\ 55-t &= \frac{55}{32} \;,\\ t &= 53\frac9{32}\;. \end{align} $

So the median corresponds to people who will live up to the age of $103\frac9{32}$, which seems rather unrealistic, but you can see from a plot of the survival function that this is about right.