# Gibbs sampler bivariate normal

Suppose that we have an observation$x = \bigl(\begin{smallmatrix} x_{1} \\ x_{2} \end{smallmatrix} \bigr)$ from a bivariate normal distribution with unknown mean $\mu = \bigl(\begin{smallmatrix} \mu_{1} \\ \mu_{2} \end{smallmatrix} \bigr)$ and covariance known $\Sigma=\bigl(\begin{smallmatrix} 1&\rho\\ \rho&1 \end{smallmatrix} \bigr)$. With a uniform prior on $\mu$, the posterior is $\mu|x\sim N(x,\Sigma)$. Prove that \begin{align} \mu_{1}|\mu_{2},x \,\,\,\, \sim \,\,\,N(x_{1}+\rho(\mu_{2}-x_{2}), 1-\rho^{2}). \end{align}

My guess : \begin{align} f(\mu_{1}|\mu_{2},x) = \frac{f(\mu_{1},\mu_{2}|x)}{f(\mu_{2},x)} \propto f(\mu_{1},\mu_{2}|x) \end{align}

After algebra the result follows. Am i correct ?

Thanks, Paul.

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