设二维随机变量(X,Y)的联合分布函数为F(X,Y)=A(B+arctanX)(C+arcY).求
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设二维随机变量(X,Y)的联合分布函数为F(X,Y)=A(B+arctanX)(C+arcY).求
(1)常数A,B,C
(2)关于X,Y的边缘分布函数
是arctanY
(1)常数A,B,C
(2)关于X,Y的边缘分布函数
是arctanY
![设二维随机变量(X,Y)的联合分布函数为F(X,Y)=A(B+arctanX)(C+arcY).求](/uploads/image/z/5850104-32-4.jpg?t=%E8%AE%BE%E4%BA%8C%E7%BB%B4%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%EF%BC%88X%2CY%EF%BC%89%E7%9A%84%E8%81%94%E5%90%88%E5%88%86%E5%B8%83%E5%87%BD%E6%95%B0%E4%B8%BAF%28X%2CY%29%3DA%28B%2BarctanX%29%28C%2BarcY%29.%E6%B1%82)
F(-∞,-∞)=A(B-π/2)(C-π/2)=0
F(-∞,+∞)=A(B-π/2)(C+π/2)=0
F(+∞,-∞)=A(B+π/2)(C-π/2)=0
F(+∞,+∞)=A(B+π/2)(C+π/2)=1
解得:A=1/π^2,B=π/2,C=π/2
f(x,y)=dF(x,y)/dxdy=1/[π^2 (1+x^2)(1+y^2)]
边缘函数
fx(x)=∫f(x,y)dy 从负无穷积分到正无穷
=1/[π(1+x^2)]
fy(y)=∫f(x,y)dx 从负无穷积分到正无穷
=1/[π(1+y^2)]
F(-∞,+∞)=A(B-π/2)(C+π/2)=0
F(+∞,-∞)=A(B+π/2)(C-π/2)=0
F(+∞,+∞)=A(B+π/2)(C+π/2)=1
解得:A=1/π^2,B=π/2,C=π/2
f(x,y)=dF(x,y)/dxdy=1/[π^2 (1+x^2)(1+y^2)]
边缘函数
fx(x)=∫f(x,y)dy 从负无穷积分到正无穷
=1/[π(1+x^2)]
fy(y)=∫f(x,y)dx 从负无穷积分到正无穷
=1/[π(1+y^2)]
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