Let \(\boldsymbol{X}=[X, Y]^{\top}\) be a random column vector with a bivariate normal distribution with expectation vector \(\boldsymbol{\mu}=[1,2]^{\top}\) and covariance matrix

\[ \boldsymbol{\Sigma}=\left[\begin{array}{ll} 1 & a \\ a & 4 \end{array}\right] \]

(a) What are the conditional distributions of \((Y \mid X=x)\) and \((X \mid Y=y)\) ? [Hint: use Theorem C.8.]

436

(b) Implement a Gibbs sampler to draw \(10^{3}\) samples from the bivariate distribution \(\mathscr{N}\left(\boldsymbol{\mu}, \sum\right)\) for \(a=0,1\), and 1.

75 , and plot the resulting samples.