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A critical region $C$ is the set of sample outcomes on which a test rejects its null hypothesis. Its power function is $\beta (\theta )={\mathbb{P}}_{\theta }(X\in C)$; its size of a statistical test is ${\sup }_{\theta \in {\Theta }_0}\beta (\theta )$. Randomized tests replace the indicator of $C$ by a rejection probability $\varphi (X)\in [0,1]$.

The Neyman-Pearson lemma states that for two simple hypotheses with densities $p_0,p_1$, the test ${\varphi }^{\ast }$ that is one where $p_1>k{p}_0$, zero where $p_1<k{p}_0$, and randomized on equality to have null rejection probability $\alpha$, maximizes power among tests of size at most $\alpha$. Here $k\ge 0$. To prove it, let $\varphi$ be any competing test. Pointwise, $({\varphi }^{\ast }-\varphi )(p_1-k{p}_0)\ge 0$, since both factors have the same sign off the equality set. Integrating givesThis proves the lemma, including equality-set randomization and points where $p_0=0$.

Take the exponential parameter to be its rate: $p_{\lambda }(x)=\lambda e^{-\lambda x}$ for $x\ge 0$. For $T={\sum }_i{X}_i$, the joint likelihood ratio is $({\lambda }_1/{\lambda }_0{)}^n{e}^{-({\lambda }_1-{\lambda }_0)T}$, decreasing in $T$. Hence for $0<\alpha <1$ the most powerful test isThere is no boundary randomization because $T$ has a continuous gamma distribution. Its density follows by repeated convolution: $p_T(t)={\lambda }^n{t}^{n-1}{e}^{-\lambda t}/(n-1)!$. Integrating by parts gives the exponential-rate likelihood-ratio test powerThe threshold is the unique positive solution of this formula at $\lambda ={\lambda }_0$ with value $\alpha$. Differentiating the finite sum yieldsThus the power function is strictly increasing, and its supremum over $0<\lambda \le {\lambda }_0$ is $\beta ({\lambda }_0)=\alpha$. Therefore the same test has size $\alpha$ for the composite null $\lambda \le {\lambda }_0$. Since the threshold does not depend on the particular larger rate, it is also uniformly most powerful against all $\lambda >{\lambda }_0$. The endpoint sizes zero and one have their trivial constant tests.