一个概率题(X,Y)服从2维正态分布 则(aX+bY)服从什么分布?
(X,Y)服从2维正态分布
对于任意a,b,(aX+bY)服从1维正态分布。
(X,Y)~N(α1,[σ1]^2,α2,[σ2]^2,r)
==>α1=E(X),[σ1]^2=D(X),
α2=E(Y),[σ2]^2=D(Y),
cov(X,Y)=rσ1σ2
==>
E(aX+bY)=aα1+bα2,
D(aX+bY)=a^2D(X)+b^2D(Y)+2abcov(X,Y)=
=a^2[σ1]^2+b^2[σ1]^2+2abrσ1σ2
==>
aX+bY~N(aα1+bα2,a^2[σ1]^2+b^2[σ1]^2+2abrσ1σ2)。全部
(X,Y)服从2维正态分布
对于任意a,b,(aX+bY)服从1维正态分布。
(X,Y)~N(α1,[σ1]^2,α2,[σ2]^2,r)
==>α1=E(X),[σ1]^2=D(X),
α2=E(Y),[σ2]^2=D(Y),
cov(X,Y)=rσ1σ2
==>
E(aX+bY)=aα1+bα2,
D(aX+bY)=a^2D(X)+b^2D(Y)+2abcov(X,Y)=
=a^2[σ1]^2+b^2[σ1]^2+2abrσ1σ2
==>
aX+bY~N(aα1+bα2,a^2[σ1]^2+b^2[σ1]^2+2abrσ1σ2)。收起