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The question is:

Let $T$ be a continuous random variable with survivor function $S$ defined on the interval $[0, \omega]$.

Now consider the random variable $S(T)$, the survivor function evaluated at the unknown lifetime value $T$.

Show that $S(T)$ has a Uniform$[0,1]$ distribution.

My attempt at answering it is:

$P(S(T) \leq x) = P(T \leq S^{-1}(x))$

Where $S^{-1}(x) = \inf\left\lbrace t : S(t) \leq x \right\rbrace$

So then

$P(T \leq S^{-1}(x)) = 1 - P(T > S^{-1}(x)) = 1 -S(S^{-1}(x)) = 1 - x$

But the cdf of a Uniform$[0,1]$ distribution should be $x$ not $1-x$?

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up vote 1 down vote accepted

Note that the survival function is one minus the cumulative distribution function, so you have

$$S(x) = 1-P(T\leq x) = P(T> x)$$

Also, since it is non-increasing, you need to reverse inequality signs when you apply its inverse to both sides of an inequality. Hence (modulo equality)

$$ \begin{align} P(S(T)\leq x) & = P(T\geq S^{-1}(x)) \\ & = S(S^{-1}(x)) \\ & = x \end{align} $$

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Thanks, out of interest, could you explain where my attempt went wrong? – Liam Williams May 23 '12 at 15:39
I edited to explain it a bit better - essentially, when you apply $S^{-1}$ to both sides of the inequality, you need to reverse the direction of the inequality, because $S$ is decreasing. – Chris Taylor May 23 '12 at 15:41
Ah I see now, thanks again. – Liam Williams May 23 '12 at 15:43

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