Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 12 / g / i / Solution

As $r_0\to +\infty$, the exponential factor tends to zero, so $P(r>r_0)\to 0$. Requiring an arbitrarily large amount of remaining time must have vanishing probability for this finite-mean distribution.