$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\sigma_v)$.

How can I design a random variable $X$ depending on $Z$, $\sigma_y$, $\sigma_v$, and $\sigma_x$ which results in $X$ having the distribution $X\sim N(0,\sigma_x)$ and having a maximal correlation with $Y$?

Initial idea: We can take $\hat{Y}$, the minimum mean square estimation (MMSE) of $Y$, and let $X=t\hat{Y}$ with $t$ chosen to ensure that $X$ has the desired variance. But later I found this intuitive idea is not correct. I wonder if someone can answer this question.