= Exponential-rate likelihood-ratio test
{title2=$\beta(\lambda)=1-e^{-\lambda c}\sum_{j=0}^{n-1}(\lambda c)^j/j!$}
For independent <exponential distribution> observations with rate $\lambda$, the likelihood ratio for a larger simple rate is decreasing in their sum. The <Neyman-Pearson lemma> therefore selects a lower-tail sum threshold $c$. The sum has a <gamma distribution>, giving the displayed power; its derivative is $c e^{-\lambda c}(\lambda c)^{n-1}/(n-1)!>0$. Calibrating at $\lambda_0$ gives size $\alpha$ for the composite null $\lambda\le\lambda_0$ and a uniformly most powerful test against larger rates.
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