= Rate function of an exponential sample mean
{title2=$I(x)=\lambda x-1-\log(\lambda x)\quad(x>0)$}
For <independent and identically distributed> <exponential random variables> of rate $\lambda>0$, the sample mean has <large-deviation speed> $n$ and the displayed <good rate function>, infinite for $x\le0$. The <cumulant-generating function> is $\log[\lambda/(\lambda-\theta)]$ for $\theta<\lambda$. Its <Legendre-Fenchel transform> has maximizing parameter $\theta=\lambda-1/x$. For $x\le0$, sending $\theta\to-\infty$ gives infinite cost. The cost has its unique zero at $1/\lambda$ and diverges at both ends of its positive domain.
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