統計學邊際分配條件分配

請問:X│Y的邊際分配,如何定義?

2010-03-12 02:05:19 補充：
性質:∫[-∞~∞] exp(-a x^2) dx =√ (π/a) , for a>0
證明:(略)

己知 f(x,y)=exp[(-2/3)(x^2/4+y^2/9-xy/6)] /(6π√3), -∞ <x,y<∞
(a) X的邊際分配f_x(x)=∫[-∞~∞] f(x,y)dy
=1/(6π√3)∫[-∞~∞] exp[(-2/27)(y-3x/4)^2]*exp(-x^2/8) dy
=1/(6π√3) exp(-x^2/8) ∫[-∞~∞] exp[(-2/27)u^2] du (u=y-3x/4)
=1/(6π√3) exp(-x^2/8) *√(27π/2)
=1/√(8π) exp(-x^2/ 8)
(b)
先求 f_y(y)=∫[-∞~∞] f(x,y)dx
=1/(6π√3)∫[-∞~∞] exp[(-1/6)(x-y/3)^2]*exp(-y^2/18) dx
=1/(6π√3) exp(-y^2/18) ∫[-∞~∞] exp[(-1/6)u^2] du (u=x-y/3)
=1/(6π√3) exp(-y^2/18) *√(6π)
=1/[3√(2π)] exp(-y^2/ 18)
f(X|Y)=f(x,y)/f_y(y)
= 1/√(6π) exp[(-1/6) (x -y/3)^2]

Note:算式很不好key in,故省略較多,請自行驗證.