Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 79 / 1 / a / Solution

At large-deviation speed $L$, the exponential distribution gives the large deviation principle on $\mathbb{R}$ with good rate functionIn particular, $\mathbb{P}(B/L\ge b)=e^{-\lambda Lb}$ for $b\ge 0$. Its finite sublevel sets are the compact sets $[0,r/\lambda ]$.