Introduction to Probability and Statistics for Engineers and Scientists

398 Chapter 9: Regression

Multiple Linear Regression

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of the X-matrix:

of the X-matrix:

FIGURE 9.15

It follows from Equation 9.10.3 that the least squares estimatorsB 0 ,B 1 ,...,Bk—
the elements of the matrixB— are all linear combinations of the independent normal
random variablesY 1 ,...,Ynand so will also be normally distributed. Indeed in such
a situation — namely, when each member of a set of random variables can be expressed
as a linear combination of independent normal random variables — we say that the set of
random variables has a jointmultivariate normal distribution.
The least squares estimators turn out to be unbiased. This can be shown as follows:

E[B]=E[(X′X)−^1 X′Y]

=E[(X′X)−^1 X′(Xβ+e)] sinceY=Xβ+e

=E[(X′X)−^1 X′Xβ+(X′X)−^1 X′e]

=E[β+(X′X)−^1 X′e]

=β+(X′X)−^1 X′E[e]

=β