Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

148 Multivariate Distributions

may not be one-to-one. Suppose, however, that we can representSas the union of

a finite number, sayk, of mutually disjoint setsA 1 ,A 2 ,...,Akso that

y 1 =u 1 (x 1 ,x 2 ,...,xn),...,yn=un(x 1 ,x 2 ,...,xn)

define a one-to-one transformation of eachAiontoT. Thus to each point inT

there corresponds exactly one point in each ofA 1 ,A 2 ,...,Ak.Fori=1,...,k,let

x 1 =w 1 i(y 1 ,y 2 ,...,yn),x 2 =w 2 i(y 1 ,y 2 ,...,yn),...,xn=wni(y 1 ,y 2 ,...,yn),

denote thekgroups ofninverse functions, one group for each of thesektransfor-
mations. Let the first partial derivatives be continuous and let each

Ji=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∂w 1 i

∂y 1

∂w 1 i

∂y 2 ···

∂w 1 i

∂yn

∂w 2 i

∂y 1

∂w 2 i

∂y 2 ···

∂w 2 i

∂yn

..

.

..

.

..

.

∂wni

∂y 1

∂wni

∂y 2 ···

∂wni

∂yn

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

,i=1, 2 ,...,k,

be not identically equal to zero inT. Considering the probability of the union

ofkmutually exclusive events and by applying the change-of-variable technique

to the probability of each of these events, it can be seen that the joint pdf of

Y 1 =u 1 (X 1 ,X 2 ,...,Xn),Y 2 =u 2 (X 1 ,X 2 ,...,Xn),...,Yn=un(X 1 ,X 2 ,...,Xn),

is given by

g(y 1 ,y 2 ,...,yn)=

∑k

i=1

f[w 1 i(y 1 ,...,yn),...,wni(y 1 ,...,yn)]|Ji|,

provided that (y 1 ,y 2 ,...,yn)∈T, and equals zero elsewhere. The pdf of anyYi,
sayY 1 ,isthen

g 1 (y 1 )=

∫∞

−∞

···

∫∞

−∞

g(y 1 ,y 2 ,...,yn)dy 2 ···dyn.

Example 2.7.3. LetX 1 andX 2 have the joint pdf defined over the unit circle
given by

f(x 1 ,x 2 )=

{ 1

π^0 <x

2

1 +x

2

2 <^1

0elsewhere.

LetY 1 =X 12 +X 22 andY 2 =X 12 /(X 12 +X 22 ). Thusy 1 y 2 =x^21 andx^22 =y 1 (1−y 2 ).

The supportSmaps ontoT ={(y 1 ,y 2 ): 0<yi< 1 ,i=1, 2 }. For each ordered

pair (y 1 ,y 2 )∈T, there are four points inS,givenby

(x 1 ,x 2 ) such that x 1 =

√

y 1 y 2 andx 2 =

√

y 1 (1−y 2 )

(x 1 ,x 2 ) such that x 1 =

√

y 1 y 2 andx 2 =−

√

y 1 (1−y 2 )

(x 1 ,x 2 ) such that x 1 =−

√

y 1 y 2 andx 2 =

√

y 1 (1−y 2 )

and (x 1 ,x 2 ) such that x 1 =−

√

y 1 y 2 andx 2 =−

√

y 1 (1−y 2 ).