Question

设二维随机变量（X，Y）的联合分布函数为F(x,y)=A(B+arctan x/2)(C+arcta

设二维随机变量（X，Y）的联合分布函数为F(x,y)=A(B+arctan x/2)(C+arctan y/3)，求常数A，B，C
(x,y)关于X和y的边缘概率密度
x和y是否相互独立

Answer 1

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)]