设随机变量x与y相互独立,X的概率分布P{X=i}=(i=-1,0,1),Y的概率密度为fY(y)=,记Z=X+Y. (Ⅰ)求P{Z≤|X=0}; (Ⅱ)求Z的概率密度fZ(z).

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问题 设随机变量x与y相互独立,X的概率分布P{X=i}=(i=-1,0,1),Y的概率密度为fY(y)=,记Z=X+Y.
    (Ⅰ)求P{Z≤|X=0};
    (Ⅱ)求Z的概率密度fZ(z).

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答案(Ⅰ)P{Z≤[*]|X=0}=P{X+Y≤[*]|X=0}=P{0+Y≤[*]|X=0} =[*] (Ⅱ)Y的分布函数为:FY(y)=[*] Z的分布函数为 FZ(z)=P{Z≤z}=P{X+y≤z}=[*]P{X+Y≤z|X=i}P{X=i} =P{-1+Y≤z|X=-1}.[*]+P{0+Y≤z|X=0}.[*]+P{1+Y≤z|X=1}.[*] =[*][P{Y≤z+1}+P{Y≤z}+P{Y≤z-1}]=[*][FY(z+1)+FY(z)+FY(z-1)] [*] 故fZ(z)=F′Z(z)=[*]

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