Ian Chen 發問時間: 科學數學 · 8 年前

我有統計學問題~急><

prove by induction by

p(E1∪ E2∪ ...∪ En)≦上nΣ下i=1p(Ei)

for any finite sequence of events E1,E2,...,and En

please step by step!

1 個解答

評分
  • yc
    Lv 5
    8 年前
    最佳解答

    Let S(n) be the statement "p(E1∪ E2∪ ...∪ En)≦Σ(i=1→n)p(Ei) where n is a positive integer.

    When n=1, p(E1∪ E2∪ ...∪ En)=p(E1), Σ(i=1→n)p(Ei)=p(E1).

    ∴S(1) is true.

    Assuming S(k) is true for some positive integer k,

    i.e. p(E1∪ E2∪ ...∪ Ek)≦Σ(i=1→k)p(Ei) .

    When n=k+1,

    p(E1∪ E2∪ ...∪ Ek+1)

    =p[(E1∪ E2∪ ...∪ Ek)∪ Ek+1]

    =p(E1∪ E2∪ ...∪ Ek)+p(Ek+1)-p[(E1∪ E2∪ ...∪ Ek)∩ Ek+1]

    ≦p(E1∪ E2∪ ...∪ Ek)+p(Ek+1)

    =Σ(i=1→k)p(Ei)+p(Ek+1)

    =Σ(i=1→k+1)p(Ei)

    ∴S(k+1) is true if S(k) is true.

    By principle of MI, S(n) is true for all positive integers n.

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