 Solution

Write $\bar{X}=n^{-1}{\sum }_i{X}_i$ and $\bar{Y}=n^{-1}{\sum }_i{Y}_i$. Completing the square in each sample, the log likelihood is, up to a data-dependent constant,The maximum-likelihood estimators follow because it is strictly maximized over unrestricted means at $(\bar{X},\bar{Y})$. Under the null it is maximized at $(\bar{X},0)$. Thus the likelihood ratio is $\Lambda =\exp (-n{\bar{Y}}^2/2)$, and the likelihood-ratio test rejects for large $T=-2\log \Lambda =n{\bar{Y}}^2$.

Under every null value of ${\mu }_X$, $Z=\sqrt{n}\bar{Y}$ has the standard Gaussian distribution; hence $T$ has the chi-squared distribution with one degree of freedom. Let $z_p={\Phi }^{-1}(p)$, where $\Phi$ is the standard Gaussian cumulative distribution function. The exact size-$\alpha$ test isThe null rejection probability is $2[1-\Phi (z_{1-\alpha /2})]=\alpha$, independently of the nuisance mean ${\mu }_X$. No randomization is required because the statistic has a continuous distribution.