563489578934

B–9 Multivariate Statistics 709
B–9 MULTIVARIATE STATISTICS
In Sec. B–3, the probabilities of simple, events, joint probabilities, and conditional probabilities
were defined. In Secs. B–4 and B–5, using the probability of simple events, we developed the concepts
of PDFs and CDFs. The PDFs and CDFs involved only one random variable, so these are
one-dimensional problems. Similarly, the moments involved only one-dimensional integration.
In this section, multiple-dimensional problems, also called multivariate statistics, will
be developed. These involve PDFs and CDFs that are associated with probabilities of intersecting
events and with conditional probabilities. In addition, multiple-dimensional moments
will be obtained as an extension of the one-dimensional moments that were studied in
Sec. B–6. If the reader clearly understands the one-dimensional case (developed in the preceding
sections), there will be little difficulty in generalizing those results to the N-dimensional case.
Multivariate CDFs and PDFs
DEFINITION.
The N-dimensional CDF is
F(a 1 , a 2 , Á , a N ) = P[(x 1 … a 1 )(x 2 … a 2 ) Á (x N … a N )]
n
lim (x1 … a 1 )(x 2 … a 2 ) Á(x N … a N )
B R
n
= n:q
(B–81)
where the notation (x 1 … a 1 )(x 2 … a 2 ) Á (x N … a N ) is the intersection event consisting
of the intersection of the events associated with x 1 a 1 , x 2 a 2 , etc.
DEFINITION.
The N-dimensional PDF is
where a and x are the row vectors, a = (a 1 , a 2 ,..., a N ), and x = f(x 1 , x 2 ,..., x N ).
DEFINITION.
f(x 1 , x 2 , Á , x N ) = 0N F(a 1 , a 2 , Á , a N )
0a 1 0a 2 Á 0a N
The expected value of y = h(x) is
`
a = x
(B–82)
* f(x 1 , x 2 , Á , x N ) dx 1 dx 2 Á dx N
(B–83)
Some properties of N-dimensional random variables are
1. f(x 1 , x 2 , Á , x N ) Ú 0
(B–84a)
q q q
2. Á f(x 1 , x 2 , Á , x N ) dx 1 dx 2
Á dxN = 1
(B–84b)
L -q L -q
-q L
[y] = h(x 1 , x 2 , Á , x N )
=
L
q
q
-q L-q
q
Á h(x 1 , x 2 , Á , x N )
L
-q