Page 221 - Probability and Statistical Inference

198 4. Functions of Random Variables and Sampling Distribution

random variables X , ..., X , with n ≥ 2, to a new set of n random variables Y ,
1
n
1
..., Y defined as follows:
n

Y , ..., Y so defined are referred to as the Helmert variables.
1 n
Let us denote the matrix

Then, one has Y = Ax where x’ = (x , ..., x ) and y’ = (y , ..., y ).
1 n 1 n
A is an orthogonal matriX. So, A’ is the inverse of A. This
n×n
implies that
n
In ℜ , a sphere in the x-coordinates continues to look
like a sphere in the y-coordinates when the x axes are rotated
orthogonally to match with the new y axes.

Observe that the matrix J defined in (4.4.3) coincides with the matrix A’ in
the present situation and hence one can immediately write | det(J) |= | det(A’)
|=| {det(AA’)} |= 1.
½
Now, the joint pdf of X , ..., X is given by
1 n
for –∞ < x , ..., x < ∞, and thus using (4.4.4) we obtain the joint pdf of Y , ...,
1
1
n
Y as follows:
n