Uniform distribution of survivor evaluated at lifetime 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 $1-x$?

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)

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