如何理解积分上限函数的导数为1？

Question

如题所述

Answer 1

解：fX(x)=∫(-∞,+∞)f(x,y)dy=3x²，0<=x<=1，其他为0
fY(y)=∫(-∞,+∞)f(x,y)dx=2y, 0<=y<=1,其他为0
(2)f(x,y)=fX(x)fY(y)
所以，x,y独立
（3）P(x>y)=∫∫(-∞,+∞)f(x,y)dxdy 积分区域为x>y
=∫(0,1)∫(0,x)f(x,y)dydx=3/5
(4)F(x,y)= ∫∫(-∞,+∞)f(x,y)dxdy ,
x<0,或者y<0,
F(x,y)=0
0<=x<=1,0<=y<=1
F(x,y)= ∫(-∞,x)∫(-∞,y)f(x,y)dxdy =x³y²
0<=x<=1,y>1
F(x,y)= ∫(-∞,x)∫(-∞,1)f(x,y)dxdy =x³
0<=y<=1,x>1
F(x,y)= ∫(-∞,1)∫(-∞,y)f(x,y)dxdy =y²
x>1,y>1
F(x,y)=1
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