563489578934

710
3. F(a 1 , a 2 ,..., a N )
Probability and Random Variables
Appendix B
= lim e:0
a 1 +e
e 7 0 L-q L-q
a 2 +e
a N +e
Á f(x 1 , x 2 , Á , x N ) dx 1 dx 2 Á dx N
L
-q
(B–84c)
4. F(a 1 , a 2 , Á , a N ) K 0 if any a i = -q, i = 1, 2, Á N
(B–84d)
5. F(a 1 , a 2 , Á , a N ) = 1 when all a i = +q, i = 1, 2, Á , N
(B–84e)
6. P[(a 1 6 x 1 … b 1 )(a 2 6 x 2 … b 2 ) Á (a N 6 x N … b N )]
b 1 +e b 2 +e b N +e
= lim
(B–84f)
e:0 L Á f(x 1 , x 2 , Á , x N ) dx 1 dx 2 Á dx N
a
e 7 0 1 +e La 2 +e L a N +e
The definitions and properties for multivariate PDFs and CDFs are based on the concepts
of joint probabilities, as discussed in Sec. B–3. In a similar way, conditional PDFs and CDFs can
be obtained [Papoulis, 1984]. Using the property P(AB) = P(A) P(BA) from Eq. (B–8), we find
that the joint PDF of x 1 and x 2 is
f(x 1 , x 2 ) = f(x 1 )f(x 2 ƒ x 1 )
(B–85)
where f(x 2 x 1 ) is the conditional PDF of x 2 given x 1 . Generalizing further, we obtain
f(x 1 , x 2 ƒ x 3 ) = f(x 1 ƒ x 3 )f(x 2 ƒ x 1 , x 3 )
(B–86)
Many other expressions for relationships between multiple-dimensional PDFs should also be
apparent. When x 1 and x 2 are independent, f(x 2 ƒ x 1 ) = f x2 (x 2 ) and
f x (x 1 , x 2 ) = f x1 (x 1 )f x2 (x 2 )
(B–87)
where the subscript x 1 denotes the PDF associated with x 1 , and the subscript x 2 denotes the
PDF associated with x 2 . For N independent random variables, this becomes
f x (x 1 , x 2 , Á , x N ) = f x1 (x 1 )f x2 (x 2 ) Á f xN (x N )
(B–88)
THEOREM. If the Nth-dimensional PDF of x is known, then the Lth-dimensional PDF
of x can be obtained when L N by
f(x 1 , x 2 , Á , x L )
q
=
L
q
-q L-q
q
Á f(x 1 , x 2 , Á , x N ) dx L+1 dx L+2 Á dx N
L
-q
(B–89)
e
N - L integrats
This Lth-dimensional PDF, where L N, is sometimes called the marginal PDF, since it is
obtained from a higher dimensional (Nth) PDF.