Conditional expectations in matrix form

c

• With random sampling:

\[E\left( y_i \right|X)=E\left( y_i \right|x_i)\]

• since \(y_i\) is independent of \(x_j\) if \(j≠i\) .
• Definition of \(E\left( y \right|X)\) :

\[E\left( y \right|X)=\begin{bmatrix}E\left( y_1 \right|X) \\ ⋮ \\ E\left( y_n \right|X)\end{bmatrix}=\begin{bmatrix}E\left( y_1 \right|x_1) \\ ⋮ \\ E\left( y_n \right|x_n)\end{bmatrix}\]

• \(E\left( y \right|X)\) is \(n×1\) .
• Result: with random sampling and a linear statistical model we have

\[E\left( y \right|X)=\begin{bmatrix}x'_1β \\ ⋮ \\ x'_nβ\end{bmatrix}=Xβ\]