推 JIVOANP:邱啥小定叮噹 10/24 17:02
※ 引述《chspfang (小汪)》之銘言：
: Prove that the inequality
: P(X≧1,Y≧1)≦min( E(X) , E(Y) )
: holds for any two non-negative continuous random variables X and Y with
: joint density f(x,y),where X is not necessarily independent of Y and min(a,b)
: equals the smaller value between a and b.
: 答案
: 由馬可夫不等式知 P(X≧1)≦E(X) 且 P(Y≧1)≦E(Y)
: P(X≧1,Y≧1)≦P(X≧1,Y≧1)+P(X≧1,Y< 1)=P(X≧1)≦E(X)
: P(X≧1,Y≧1)≦P(X≧1,Y≧1)+P(X <1,Y≧1)=P(Y≧1)≦E(Y)
: => P(X≧1,Y≧1)≦min( E(X),E(Y) )
: 答案大概是這樣
: 可是解答我看不太懂
: 有強者能出來解釋一下嗎
P(X≧1,Y≧1)+P(X≧1,Y< 1)=P(X≧1)
P(X≧1,Y≧1)+P(X <1,Y≧1)=P(Y≧1) 這個應該沒問題
P(X≧1,Y≧1)≦P(X≧1)≦E(X)且P(X≧1,Y≧1)≦P(Y≧1)≦E(Y)
所以P(X≧1,Y≧1)小於E(X),E(Y)中的最小值
=> P(X≧1,Y≧1)≦min( E(X),E(Y) )
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