Given figure shows the data along with a fitted quadratic

Question

Still in connection with given Example, show that the posterior distribution on (a, b, c, σ ^{- 2}) is a standard distribution made of a trivariate normal on (a, b, c) conditional on σ and the data and a gamma distribution on σ ^-2 given the data.

Example

The cars dataset relates braking distance (y) to speed (x) in a sample of cars. Given Figure shows the data along with a fitted quadratic curve that is given by the R function lm. The model posited for this dataset is a quadratic model

y_ij = a + bx_i + cx_i² + ε_ij,    i = 1,...,k,      j = 1,...n_i,

where we assume that εij ∼ N(0, σ²) and independent.