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4. Functions of Random Variables and Sampling Distribution 195

4.4.1 Several Variable Situations
In this subsection, we address the case of several variables. We start with the
machinery available for one-to-one transformations in a general case. Later,
we include an example when the transformation is not one-to-one. First, we
clarify the techniques in the two-dimensional case. One may review the Sec-
tion 4.8 for some of the details about matrices.
Suppose that we start with real valued random variables X , ..., X and we
n
1
have the transformed real valued random variables Y = g (X , ..., X ), i = 1, ...,
i
1
n
i
n where the transformation from (x , ..., x ) ∈ χ to (y , ..., y ), ∈ is assumed
n
1
n
1
to be one-to one. For example, with n = 3, we may have on hand three
transformed random variables Y = X + X , Y = X – X , and Y = X + X +
1
1
2
2
1
3
2
2
1
X .
3
We start with the joint pdf f(x , ..., x ) of X , ..., X and first replace all the
1
1
n
n
variables x , ..., x appearing within the expression of f(x , ..., x ) in terms of
1
n
1
n
the transformed variables i = 1, ..., n. In other words, since
the transformation (x , ..., x ) to (y , ..., y ) is one-to one, we can theoreticallY
n
1
n
1
think of uniquely expressing x in terms of (y , ..., y ) and write x = b (y , ...,
1
n
i
i
1
i
y ), i = 1, ..., n. Thus, f(x , ..., x ) will be replaced by f(b , ..., b ). Then, we
n
1
n
n
1
multiply f(b , ..., b ) by the absolute value of the determinant of the Jacobian
n
1
matrix of transformation, written exclusively involving the transformed vari-
ables y , ..., y only. Let us define the matrix
1 n
The understanding is that each x variable is replaced by b (y , ..., y ), i = 1, ...,
1
i
i
n
n, while forming the matrix J. Let det(J) stand for the determinant of the
matrix J and | det(J) | stand for the absolute value of det(J). Then the joint pdf
of the transformed random variables Y , ..., Y is given by
1 n
for y ∈ γ.
On the surface, this approach may appear complicated but actually it is not
quite so. The steps involved are explained by means of couple of examples.
Example 4.4.5 Let X , X be independent random variables and X be
1
2
i
distributed as Gamma(α , β), α > 0, β > 0, i = 1, 2. Define the trans-
i
i
formed variables Y = X + X , Y = X /(X + X ). Then one can uniquely
1 1 2 2 1 1 2