Cross Product Sample Spaces
495
where w ranges over the outcomes (W0, w2) in 92. By definition of p, the left-hand side is
YE
(pl((0l)'p2((02))
(W1l402) GO
This expression represents the sum of all possible products Pi (l0) • p2 (o02). The product
of F., p1(wo) with ZY-Ž2 P2(w), however, is the same quantity as the sum of all possible
products p I(w) • P2(w) by the Product of Sums Principle. Also, P1 and P2 are probability
densities on Q
2 , and 922, respectively, so
Y pl(w)=
P2((02)
1
wIEl
2 1
W2EQ2
Hence,
Pw•• Ip (Ogl) - P2 (62)) =
llpl
(oil))
,•
P2((092))
-1.1
This completes the proof.
U
8.3.2 The Cross Product of Sample Spaces
The sample space Q2 of Theorem 1, which is endowed with the probability density function p given by that theorem, is called a cross product sample space. This notation was
introduced in Section 1.3.4, but we recall it for convenience here. Such a sample space 92
is usually denoted 921 X 022 to indicate that it is based on the two sample spaces Q1 and
Q22. In fact, we can form a cross product sample space from any number n > 2 of sample spaces. This construction involves taking a cross product of sets, as defined below in
Definition 1, and then endowing this new set with a certain probability density, as given in
Definition 2.
Definition 1. The cross product of n sets
S 1 ,S 2 .
an
is the set of all n-tuples
(s1, S2.
S)
where Si E Si for I Notation. The cross product of n sets is denoted as S1 x S2 x ... x Sn. When the sets
are all copies of the same set S, then the cross product is denoted Sn.
The cross product of a countably infinite ordered list of sets
S1,$2 ....
is the set of all sequences
(sl,s2 .... )