Design a random variable which has the maximal correlation with another random variable $Y$ 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.
 A: It can be obtained that the MMSE of $Y$ is given by $\hat{Y}=\frac{\sigma_y}{\sigma_y+\sigma_v}Z=kZ$, and the covariance of $\hat{Y}$ is $k^2(\sigma_y+\sigma_v)$. Now suppose $X=t\hat{Y}$, to ensure $X$ has covariance $\sigma_x$, we get
$$t^2k^2(\sigma_y+\sigma_v)=\sigma_x$$
Solving $t$ and substituting into $X$ leads to
$$X=\sqrt{\frac{\sigma_x}{k^2(\sigma_y+\sigma_v)}}\hat{Y}=\sqrt{\frac{\sigma_x}{k^2(\sigma_y+\sigma_v)}}kZ=\sqrt{\frac{\sigma_x}{\sigma_y+\sigma_v}}Z$$
It seems that $X$ has nothing to do with the MMSE of $Y$. We just define a new variable
$$\bar{t}=\sqrt{\frac{\sigma_x}{\sigma_y+\sigma_v}}$$
Then $X$ can be derived directly by
$$X=\bar{t}Z$$
We can also calculate the expectation of $XY$, i.e.,
$$\mathbb{E}[XY]=\bar{t}\mathbb{E}[ZY]=\bar{t}\sigma_y$$
My intuitive idea is that: If we know the exact value of $Y$, we can simply set $X=\frac{\sigma_x}{\sigma_y}Y$, which ensure maximal correlation between $X$ and $Y$. But now, since we cannot know the exact value of $Y$, we use its MMSE estimation. Now the above derivation states that we can simple use the corrupted measurement $Z$, to design $X$. I wonder if there is any in deep explanation of this problem.
