Re: [機統] P(X=c)=I if and only if VAR(X)=0

看板Math作者THEJOY (最後的演武)時間14年前 (2011/11/25 17:39)推噓1(1推 0噓 6→)

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※ 引述《touurtn (vv)》之銘言： : 請問 : 在證到 P(|X-EX|<ε)=1 : 方法1： : 有些書會寫：since ε is arbitrary, then P( X=EX )=1 : 方法2： : 但我使用的課本(A Courese in Mathematical Statistics ,Roussas) : 卻還要利用到 P(lim An) = lim p(An)這件事 : 老師的證明手法也是採用這樣，為什麼會分成這兩種呢？ Since P(|X-EX|<ε)=1 for ε>0, so it is true for ε=1/n>0,n=1,2,... Let A_n={|X-EX|<1/n}, then P(A_n)=1 for all n=1,2,... by assumpion. And obviously,{A_n} forms a decreasing sequence and A_n->{|X-EX|=0} as n->00 So by continuity of measure, P(X=EX) = P(|X-EX|=0) = P(lim A_n) = P(∩A_n) = limP(A_n) = 1 n n n In fact, if 0≦|X-EX|<ε is true for all ε>0, then |X-EX|=0 in the sense of limit. -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.119.246.32

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