推 jshk:感謝分享 09/06 20:17
※ 引述《lai90043 (賴伯)》之銘言：
: 1.consider a nonnegative integer valued random variable X
: suppose that E[X] = 10 & P(X=0) = 0.1 find E[max(0,X-1)]
幫解第一題
0 ; if X=0,1
首先討論max(0,X-1)= 其中X為非負整數
X-1; if X=2,3,4,...
=> E[max(0,X-1)]=0*p[X=0]+0*p[X=1]+p[X=2]+2p[X=3]+....
=p[X=2]+2p[X=3]+.... -----(1)式
接著思考題目所給線索
1.E[X]= 10 =0*p[X=0]+p[X=1]+2p[X=2]+3p[X=3]+...
=p[X=1]+2p[X=2]+3p[X=3]+....-----(2)式
2.p[X=0] = 0.1 = 1-{p[X=1]+p[X=2]+p[X=3]+...}
=>p[X=1]+p[X=2]+p[X=3]+... = 0.9 -----(3)式
3.(2)式-(1)式 => 10-E[max(0,X-1)] = p[X=1]+p[X=2]+p[X=3]+...
將(3)式結果代入得 E[max(0,X-1)] = 10-0.9 = 9.1
^^^^ANS
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