8.1 Regression coefficients

We can write the regression coefficients as

where $A={(X^{\prime }X)}^{-1}{X}^{\prime }$.

Since $A$ is considered to be fixed, $\widehat{\beta }$ is a linear combination of the random variables $Y_1,\mathrm{\dots },Y_n$. By the definition of the linear model $Y_1,\mathrm{\dots },Y_n$ are normal random variables, and so any linear combination of $Y_1,\mathrm{\dots },Y_n$ is also a normal random variable (by the linearity property of the normal distribution, see Math230).