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2
Mathematical Preliminaries
2.3.3 Conditional Densities
Consider two events A and B. As stated earlier, their intersection Α η Β is
also an event. The conditional probability of A, given that Β has occurred,
is defined by
»{A
I
Β) = &{A η B)I^(B) (71)
The conditional distribution P
f
{z\B) of a random variable f, assuming Β has
occurred, is defined as the conditional probability of the event {f ^ z};
that is,
P
f
{z\B)
= 0>{f^z\B} = 9{{t ^ ζ) η B)\9(B) (72)
where the numerator is the event consisting of all outcomes ω, such that
f (ω,) ^ ζ and ω
ί
e Β (73)
At the points of continuity of P
f
(z
\
B), the conditional density is defined as
p
f
(z\B) ...