 Solution

The Neyman-Pearson lemma states that a test which rejects where the joint likelihood ratio $L_1/L_0$ is largest is a most powerful test of its size: its power is at least that of any test with no larger size. More precisely, choose a threshold $k$ and reject if $L_1/L_0>k$, accept if it is smaller, and if necessary randomize on equality to attain the specified size.

For an independent and identically distributed statistical sample, takeThe $Y_i$ are independent and identically distributed random variables under either simple hypothesis, and the log joint ratio is their sum. Thus the nonrandomized threshold form can be writtenwith $c=-\log k$, using either strict or non-strict inequality when the boundary has probability zero. The usual finite-log expression assumes positive common support; zero densities can instead be handled by the extended likelihood ratio on the relevant supports. Continuous observations alone do not ensure a continuous likelihood-ratio statistic: ties may require the boundary randomization in the lemma. This qualification preserves the exact-size formulation.