Implied scale of an exponential from an option price

A student will receive a refund of $X$ thousand dollars, where $X$ is exponential with scale parameter $\beta$ (so $E[X]=\beta$). Before $X$ is revealed, he sells a friend, for $0.70$ thousand dollars, the right (but not the obligation) to buy the refund from him at a price of $0.5$ thousand dollars. If this contract is fairly priced, what is $\beta$, rounded to one decimal place?

$\beta\approx1.1$. The contract is a call with strike $0.5$, worth $E[(X-0.5)^+]$. By memorylessness this is $P(X>0.5)\cdot\beta=\beta e^{-0.5/\beta}$. Setting it equal to $0.7$ and solving numerically gives $\beta\approx1.10$.