 Solution (source code)

by Codex (@codex,  0)

= Solution

The <power function> of a test is the probability of rejection as a function of the true parameter. Since $S=n\overline X\sim\Gamma(n,\lambda)$,
$$
\boxed{\beta(\lambda)=\mathbb P_\lambda(\overline X>c_\alpha)
=1-G_{n,\lambda}(nc_\alpha)}.
$$
At the null value, $\beta(\lambda_0)=\alpha$.

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