= Solution
{c}
Let $S_{xx}=\sum_i x_i^2$, $S_{xy}=\sum_i x_iy_i$, and $\bar y=n^{-1}\sum_i y_i$. The independent observations in the <normal linear model> give
$$
L(\alpha,\beta)\propto
\exp\left\{-\frac1{2\sigma^2}\sum_i(y_i-\alpha-\beta x_i)^2\right\}.
$$
The centered design condition $\sum_i x_i=0$ removes the $\alpha\beta$ term:
$$
\sum_i(y_i-\alpha-\beta x_i)^2
=\sum_i y_i^2+n\alpha^2+S_{xx}\beta^2
-2n\bar y\alpha-2S_{xy}\beta.
$$
The stated <prior density> factors into independent <normal distributions> for the <regression intercept> $\alpha$ and <regression coefficient> $\beta$. Multiplication by the <likelihood function> therefore leaves separate quadratic expressions in the two coefficients. Define the positive posterior <variances>
$$
v_\alpha=\left(\frac n{\sigma^2}+\frac1{\tau_0^2}\right)^{-1},\qquad
v_\beta=\left(\frac{S_{xx}}{\sigma^2}+\frac1{\tau_1^2}\right)^{-1},
$$
and the <posterior means>
$$
m_\alpha=v_\alpha\left(\frac{n\bar y}{\sigma^2}+\frac{\alpha_0}{\tau_0^2}\right),\qquad
m_\beta=v_\beta\left(\frac{S_{xy}}{\sigma^2}+\frac{\beta_0}{\tau_1^2}\right).
$$
By <completing the square>, the joint <posterior density> is
$$
\boxed{\pi(\alpha,\beta\mid\boldsymbol y)
=\frac1{2\pi\sqrt{v_\alpha v_\beta}}
\exp\left\{-\frac{(\alpha-m_\alpha)^2}{2v_\alpha}
-\frac{(\beta-m_\beta)^2}{2v_\beta}\right\}}.
$$
Equivalently,
$$
\boxed{\alpha\mid\boldsymbol y\sim N(m_\alpha,v_\alpha),\qquad
\beta\mid\boldsymbol y\sim N(m_\beta,v_\beta),\qquad
\alpha\perp\beta\mid\boldsymbol y}.
$$
This is <Gaussian conjugacy for a normal linear model>. Each posterior <precision parameter> is the sum of its data and prior <precision parameters>. The formulas remain valid when $S_{xx}=0$: all $x_i$ are then zero, so the <posterior distribution> of $\beta$ equals its <prior distribution>.
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