: 99題目
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: 99參考解答
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第五題
f(x,y) = exp(-y)/y 0<x<y, 0<y<∞
E(X^3|y=2) = ∫x^3*f(x|y=2)dx
f(x|y) = f(x,y)/f(y) = f(x,y)/∫f(x,y)dx
又∫f(x,y)dx
=∫_{0,y} exp(-y)/y dx
= [exp(-y)/y]*∫_{0,y} 1 dx
= [exp(-y)/y]*y = exp(-y)
f(x|y) = f(x,y)/∫f(x,y)dx
= [exp(-y)/y]/[exp(-y)]
= 1/y 0<x<y
f(x|y=2) = 1/2, 0<x<2
E(X^3|y=2)
= ∫x^3*f(x|y=2)dx
= ∫_{0,2}x^3*(1/2)dx
= (1/2)*x^4/4|_{0,2}
= 2