作业帮二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)],x>=0,y>=0 =0,其它 求概率P{Y
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![二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)],x>=0,y>=0 =0,其它 求概率P{Y](/uploads/image/z/519414-6-4.jpg?t=%E4%BA%8C%E7%BB%B4%E9%9A%8F%E6%9C%BA%E5%90%91%E9%87%8F%28X%2CY%29%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%E5%87%BD%E6%95%B0%E4%B8%BA+f%28x%2Cy%29%3D2e%5E%5B-%282x%2By%29%5D%2Cx%3E%3D0%2Cy%3E%3D0+%3D0%2C%E5%85%B6%E5%AE%83+%E6%B1%82%E6%A6%82%E7%8E%87P%7BY)
二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)],x>=0,y>=0 =0,其它 求概率P{Y
二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)],x>=0,y>=0 =0,其它 求概率P{Y
二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)],x>=0,y>=0 =0,其它 求概率P{Y
P{Y≤X}=∫∫f(x,y)dxdy
y≤x
=∫(0--+∝)dx∫(0-->x)2e^[-(2x+y)]dy
=∫(0--+∝)dx∫(0-->x)2e^[-2x-y)]dy
=2∫(0--+∝)e^(-2x)dx∫(0-->x)e^(-y)]dy
=2∫(0--+∝)e^(-2x)[1-e^(-x)]dx
=2∫(0--+∝)e^(-2x)-e^(-3x)]dx
=1-2/3
=1/3
二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)]，x>=0,y>=0 =0,其它
于是P{Y<=X}=∫∫{Y<=X}f(x,y)dxdy=∫(-∞,+∞)dx∫(-∞,x)f(x,y)dy
=∫(-∞,0)dx∫(-∞,x)f(x,y)dy+∫[0,+∞)dx∫(-∞,x)f(x,y)dy
=∫(0,+∞)dx∫(-∞,x)f(x,y)dy
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二维随机向量(X,Y)概率密度函数为 f(x,y)=2e^[-(2x+y)]，x>=0,y>=0 =0,其它
于是P{Y<=X}=∫∫{Y<=X}f(x,y)dxdy=∫(-∞,+∞)dx∫(-∞,x)f(x,y)dy
=∫(-∞,0)dx∫(-∞,x)f(x,y)dy+∫[0,+∞)dx∫(-∞,x)f(x,y)dy
=∫(0,+∞)dx∫(-∞,x)f(x,y)dy
=∫(0,+∞)[∫(-∞,0)f(x,y)dy+∫[0,x)f(x,y)dy]dx
=∫(0,+∞)dx∫[0,x)f(x,y)dy
=∫(0,+∞)dx∫[0,x)2e^[-(2x+y)]dy
=-∫(0,+∞)2e^[-(2x+y)]|[0,x)dx
=∫(0,+∞)[2e^(-2x)-2e^(-3x)]dx
=-e^(-2x)|(0,+∞)+(2/3)e^(-3x)|(0,+∞)
=1-(2/3)
=1/3
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