Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 27L / c / Solution

Let $F$ be the distribution function of ${\xi }_1$, write $\overline{F}=1-F$, and fix $x>0$. SetConditioning on the first interarrival givesIndeed, if ${\xi }_1>t$, the first interval contains $t$ and must also have length at least $x$; if ${\xi }_1\le t$, the process starts afresh at time ${\xi }_1$.

Put $h=\overline{F}(x)$ and $d(t)=q(t)-h$. ThenwhereIn particular $g\ge 0$. Iterating the renewal equation yieldsFor fixed $t$, the last term tends to zero in absolute value because $d$ is bounded andIt follows thatThus the stochastic length bias of a renewal interval givesfor every $x,t>0$.