prob.part15.31_32 - 1.2. DISCRETE PROBABILITY DISTRIBUTIONS...

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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.
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prob.part15.31_32 - 1.2. DISCRETE PROBABILITY DISTRIBUTIONS...

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