Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 38 / 3 / a / Solution

For a unit-rate exponential distribution, $\mu =1$ and $M_X(r)=1/(1-r)$ for $r<1$. The adjustment coefficient equation isIts nonzero root isIt lies in $(0,1)$ for $\theta >0$, so the moment-generating function is finite there and the Lundberg inequality applies.