# 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.

• What's wrong with the intuitive idea? Jan 18 at 3:52
• @MattF. Thanks for help editing the question. Please see my answer for the intuitive idea. Jan 18 at 19:46

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.
• I think it’s right that a rescaled version of $Z$ has the highest correlation. Jan 18 at 22:59
• I made a mistake in the above derivation. The covariance of $\hat{Y}$ (MMSE of $Y$), is given by $$Cov[\hat{Y}]=\frac{\sigma_y\sigma_v}{\sigma_y+\sigma_v}$$ Jan 20 at 16:01