Suppose $X$ and $y$ are jointly Gaussian random variables with correlation coefficient $ ho$. Let $X \sim N\left(\mu_{X}, \sigma_{X}^{2} ight) $ and $Y \sim N\left(\mu_{Y}, \sigma_{Y}^{2} ight) $ Let $z=E[X \mid Y]$, show that (a) $2=\mu_{X}+\frac{ \sigma_{X}}{\sigma_{Y}} ho\left(Y-\mu_{Y} ight)$. Hint: For bivariate Gaussian random vector $(X, Y)$ with parameters given as above, the joint pdf is $$ f_{X, Y}(x, y)=\frac{\exp \left\{-\frac{1}{2\left(1- ho^{2} ight)}\left|\frac{\left(x-\mu_{X} ight)^{2}} {\sigma_{X}^{2}}+\frac{\left(y-\mu_{Y} ight)^{2}} {\sigma_{Y}^{2}}-\frac{2 ho\left(x- \mu_{X} ight)\left(y-\mu_{Y} ight)}{\sigma_{X} \sigma_{Y}} ight] ight\}}{2 \pi \sigma_{X} \sigma_{Y} \sqrt{1- ho^{2}}} $$ (b) From part (a) conclude that $E[X \mid Y]$ is a Gaussian random variable with mean $\mu_{X}$ and variance $\sigma_{X}^{2} ho^{2}$. (c) We know that $2$ is the MMSE estimator of $X$ from $y$. Find the Mean Square Error of this estimator, i.e., $E\left [CX-)^{2} ight ]$. SP.PC. 063