1.2. DISCRETE PROBABILITY DISTRIBUTIONS
23
Suppose next that
A
and
B
are disjoint subsets of Ω. Then every element
ω
of
A
∪
B
lies either in
A
and not in
B
or in
B
and not in
A
. It follows that
P
(
A
∪
B
)
=
∑
ω
∈
A
∪
B
m
(
ω
) =
∑
ω
∈
A
m
(
ω
) +
∑
ω
∈
B
m
(
ω
)
=
P
(
A
) +
P
(
B
)
,
and Property 4 is proved.
Finally, to prove Property 5, consider the disjoint union
Ω =
A
∪
˜
A .
Since
P
(Ω) = 1, the property of disjoint additivity (Property 4) implies that
1 =
P
(
A
) +
P
(
˜
A
)
,
whence
P
(
˜
A
) = 1

P
(
A
).
±
It is important to realize that Property 4 in Theorem 1.1 can be extended to
more than two sets. The general ±nite additivity property is given by the following
theorem.
Theorem 1.2
If
A
1
, . . . ,
A
n
are pairwise disjoint subsets of Ω (i.e., no two of the
A
i
’s have an element in common), then
P
(
A
1
∪ ··· ∪
A
n
) =
n
±
i
=1
P
(
A
i
)
.
Proof.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 scf
 Probability theory, Discrete probability distribution, Basic concepts in set theory, Disjoint sets

Click to edit the document details