The bivariate random variable ( Y , X ) ( Y , X ) has the following mean vector and covariance matrix: E [ X

The bivariate random variable $(Y,X)$ has the following mean vector and covariance matrix:

$E\left[\begin{array}{c}X\\ Y\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$ and $\mathrm{Cov}(X,Y)=\left[\begin{array}{cc}5& 2\\ 2& 2\end{array}\right]$

(a) Derive the values of $a$ and $b$ in $\hat{Y}=a+bX$ that minimize the expected squared distance between $Y$ and $\hat{Y}$, i.e., that produce the best linear predictor of $Y$ outcomes in terms of $X$ outcomes.

(b) What proportion of the variance in $Y$ is explained by the best linear predictor that you derived above?