The conditional distribution of y given x

Problem

Setup: a linear regression model with a random sample

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+ \ldots +β_kx_{i,k}+ε_i\]

Suppose that \(ε_i|x_i ~ N\left( 0,σ^2 \right)\) . Find the distribution of \(y_i|x_i\) .