[機統] Boole's Inequality 的證明
Boole's Inequality:
∞ ∞
P( ∪ A ) ≦Σ P(A )
i=1 i i=1 i
原本覺得可用數學歸納法證明,
可是看了Ross 的 A first course in probability,
他是原先的聯集拆成等價且互斥的集合去作證明
可是如果像這樣用數學歸納法證明:
Basis Step: P(A ) ≦ P(A )
1 1
Inductive Step:
Suppose
n n
P( ∪ A ) ≦Σ P(A )
i=1 i i=1 i
Then,
n+1 n n n
P( ∪ A ) = P(∪ A ∪ A ) = Σ P(A ) + P(A ) - P((∪ A ∩ A ))
i=1 i i=1 i n+1 i=1 i n+1 i=1 i n+1
n+1
≦ Σ P(A )
i=1 i
By induction,
n n
P( ∪ A ) ≦Σ P(A ) for all n
i=1 i i=1 i
好像有一些地方很奇怪, 但我又說不出來...
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◆ From: 114.47.141.82
※ 編輯: yueayase 來自: 114.47.141.82 (09/30 21:38)
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