
18 Modeling and In verse Problems in the Presence of Uncertainty
Definition 2.2.6 implies that if X and Y are independent, then we have
ρ
X|Y
(x|y) = p
X
(x). Hence, by (2.28), we obtain
p
XY
(x, y) = p
X
(x)p
Y
(y), (2.29)
which is also used as a definition for the independence of two random variables.
Thus, we see that if two random variables are independent, then the joint
probability de nsity function can be determined by the marginal probability
density functions. It should be noted that the definition for the independence
of two general (either discrete or continuous) random variables is based on
the independence of two events, and is given as follows.
Definition ...