Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 132

for X .

Independence of quadratic forms is the subject of Cochran Theorem, employed
to determine the F distribution.

Normal( ʼ , ʣ ), for a square invertible matrix ʣ

*

Cochran Theorem
For X ) T Normal( ʼ ,I nXn ), let Yi i =(X - ʼ ʣ i (X - ʼ ) for ʣ i symmetric idempotent

*

matrices of ranks ri i such that P n
i 1

n and P i 1 P i ¼

r i ¼ I nXn . Then the Yi are

2 r i random variables.
The distribution F is the distribution of a quotient:

independent @

F F m ; n $ F ¼ ð X 1 = m Þ= ð X 2 = n Þ

2
m

2
m

for independent X 1 @ and X 2 @ .

A.8 Regression

The process of approximating the distribution of a random variable by a conditional
distribution is known as regression.
For normal distributions, the conditional expectation of Y conditional on X is a
linear function of X.
In the case of one-dimensional X, this linear regression function is given by

ðÞþr 1 EY ð j X ¼ x Þ ¼ EY

X r Y q XY x

ð EX ðÞ Þ;

2
Y

so that the variance of the conditional expectation is .

Also, in the normal case, the conditional variance does not depend on X and is
given, for one-dimensional X, by (1

q XY r

2
XY

2
Y

q ) r

.

Starting from a sample ((Y 1 ,X 1 ), … ,(Y n ,X n )) of n independent random vectors,

the coef cients of this linear function can be approximated by those of the linear

projection operator in the sense that, for X = (X 1 , ,X n ) T and Y = (Y 1 , ,Y n ) T ,

… …

1 X T Y 1 X T r

Y

XX T X XX T X

2
Y

¼ Normal E Y ð j X Þ; :