设随机变量X,Y相互独立,且X~P(1),Y~P(2),求P{max(X,Y)≠0}及P{min(X,Y)≠0}.

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问题 设随机变量X,Y相互独立,且X~P(1),Y~P(2),求P{max(X,Y)≠0}及P{min(X,Y)≠0}.

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答案P{max(X,Y}≠0)=1一P{max(X,Y)=0}=1一P(X=0,Y=0) =1一P(X=0)P(Y=0)=1一e-1e-2=1一e-3。 P{min(X,Y)≠0}=1一P{min(X,Y)=0}, 令A={X=0},B={Y=0},则{min(X,Y)=0}=A+B, 于是P{min(X,Y)=0}=P(A+B)=P(A)+P(B)一P(AB) =e-1+e-2一e-1.e-2=e-1+e-2一e-3, 故P{min(X,Y)≠0}=1一e-1一e-2+e-3

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