Page 223 - Elements of Distribution Theory

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052184472Xc07 CUNY148/Severini May 24, 2005 3:59

7.3 Functions of a Random Vector 209

Theorem 7.3. Let X denote a random vector with an absolutely continuous distribution
d
with density function p X . Let X 1 , ··· , X m denote disjoint open subsets of R such that

m

Pr X ∈ X i = 1.
i=1
Let g denote a function on X and let g (i) denote the restriction of g to X i . Assume that, for
each i = 1,..., m, g (i) is one-to-one and continuously differentiable with inverse h (i) and
(i)
the Jacobian g is nonzero on X i . Then Y = g(X) has an absolutely continuous distribution
with density function p Y , given by
m (i)

(i) ∂h (y) m

p Y (y) = p X h (y) I {y∈Y i } , y ∈ g ∪ i=1 X i
∂y
i=1
(i)
where Y i = g (X i ).
Proof. Let f denote a bounded function on Y. Then,

m

(i)
E[ f (Y)] = f (g(x))p X (x) dx = f g (x) p X (x) dx.
X i=1 X i
On X i , g (i) is one-to-one and continuously differentiable so that the change-of-variable
formula may be applied to the integral over X i . Hence,
(i)
m

(i) ∂h (y)

E[ f (Y)] = f (y)p X h (y) dy.
∂y
i=1 Y i
The result follows by interchanging integration and summation, which is valid since the
sum is finite.
Example 7.12 (Product and ratio of standard normal random variables). Let X 1 , X 2
denote independent random variables, each with a standard normal distribution. Let
X 1
Y 1 = X 1 X 2 and Y 2 = .
X 2
Writing X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ), it follows that Y = g(X), g = (g 1 , g 2 ), where
g 1 (x) = x 1 x 2 and g 2 (x) = x 1 /x 2 .
2
Clearly this is not a one-to-one function on R ; for instance, (x 1 , x 2 ) and (−x 1 , −x 2 ) yield
the same value of g(x), as do (x 1 , −x 2 ) and (−x 1 , x 2 ).
2
+
+
The function is one-to-one on the four quadrants of R . Hence, take X 1 = R × R ,
X 2 = R × R , X 3 = R × R , and X 4 = R × R . The restriction of g to each X i is
+
−
−
−
+
−
one-to-one, with inverses given by
√ √ (2) √ √
(1)
h (y) = ( (y 1 y 2 ), (y 1 /y 2 )), h (y) = ( (y 1 y 2 ), − (y 1 /y 2 )),
√ √ (4) √ √
(3)
h (y) = (− (y 1 y 2 ), (y 1 /y 2 )), h (y) = (− (y 1 y 2 ), − (y 1 /y 2 ))
and Jacobians
(i)
1

∂h (y)
= .
∂y 2|y 2 |