2.40 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:(1)系数A,B及C;(2)(X,Y)的联合概率密度;(3)边缘分布函数及边缘概率密度.随机变量X与y是否独立?
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![2.40 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:(1)系数A,B及C;(2)(X,Y)的联合概率密度;(3)边缘分布函数及边缘概率密度.随机变量X与y是否独立?](/uploads/image/z/10208541-21-1.jpg?t=2.40+%E8%AE%BE%E4%BA%8C%E7%BB%B4%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%28X%2CY%29%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%2Barctan+x%2F2%29%28C%2Barctan+Y%2F3%29+%E6%B1%82%EF%BC%9A%281%29%E7%B3%BB%E6%95%B0A%2CB%E5%8F%8AC%EF%BC%9B%282%29%28X%2CY%29%E7%9A%84%E8%81%94%E5%90%88%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%EF%BC%9B%283%29%E8%BE%B9%E7%BC%98%E5%88%86%E5%B8%83%E5%87%BD%E6%95%B0%E5%8F%8A%E8%BE%B9%E7%BC%98%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6.%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8FX%E4%B8%8Ey%E6%98%AF%E5%90%A6%E7%8B%AC%E7%AB%8B%3F)
2.40 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:(1)系数A,B及C;(2)(X,Y)的联合概率密度;(3)边缘分布函数及边缘概率密度.随机变量X与y是否独立?
2.40 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:(1)系数A,B及C;(2)(X,Y)的联合概率密度;(3)边缘分布函数及边缘概率密度.随机变量X与y是否独立?
2.40 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:(1)系数A,B及C;(2)(X,Y)的联合概率密度;(3)边缘分布函数及边缘概率密度.随机变量X与y是否独立?
由性质得:
F(+∞,+∞)=1,
则
A(B+arctan x/2)(C+arctan Y/3) =A(B+π/2)(C+π/3)
F(-∞,+∞)=0
A(B+arctan x/2)(C+arctan Y/3) =A(B-π/2)(C+π/3)
F(+∞,-∞)=0
A(B+arctan x/2)(C+arctan Y/3) =A(B+π/2)(C-π/2)
解得:A=6/(11π),B=π/2,C=π/2
(X,Y)的联合概率密度:
6/(11π)(π/2+arctan x/2)(π/2+arctan Y/3)
边缘分布函数及边缘概率密度:
f(x)=∫f(x,y)dy
f(y)=∫f(x,y)dx
f(x,y)=d^2(F(x,y))/dxdy
所以f(x)=d(F(x,y))/dx=6/(11π)*2/(x^2+4)*(π/2+arctan Y/3)
f(y)=d(F(x,y))/dy=6/(11π)*(π/2+arctan x/2)*3/(x^2+9)