回答于 2013/09/02 21:13
让我试着推导如下:
令 P=w1*X1+w2*X2,简称X1标准差为S1,X2标准差为S2,X1期望为u1, X2期望为u2, P标准差为Sp,而方差为Vp=(Sp)^2, E为期望符号
按方差的定义: Vp = E(P^2)-E(P)^2 ---------------(1)
而 E(P^2)=E((w1*X1+w2*X2)^2) = E(w1^2*X1^2+w2^2*X2^2+2*w1*w2*X1*X2)
= w1^2*E(X1^2)+w2^2* E(X2^2)+2w1*w2*E(X1*X2) --------------------------------(2)
注意: 按方差的定义 s1^2 = E(X1^2)-u1^2所以:E(X1^2)=s1^2 + u1^2,同理:E(X2^2)=s2^2 + u2^2
代入(2)式得:E(P^2)=w1^2*(s1^2 + u1^2)+w2^2*(s2^2 + u2^2)+2w1*w2*E(X1*X2)-------------(3)
而 E(P)^2 = (E(w1*X1+w2*X2))^2 = (w1*u1+w2*u2)^2= w1^2*u1^2+w2^2*u2^2+2*w1*w2*u1*u2 ---------------(4)
将(3)和(4)式代入(1)式得:Vp = w1^2*(s1^2 + u1^2)+w2^2*(s2^2 + u2^2)+2w1*w2*E(X1*X2) - (w1^2*u1^2+w2^2*u2^2+2*w1*w2*u1*u2)
得:Vp= w1^2*s1^2 + w2^2*s2^2 + 2*w1*w2*(E(X1*X2)-u1*u2) ----------------(5)
注意:由相关系数定义 Rho(x1,x2)=(E(X1*X2)-E(x1)*E(x2))/s1/s2, 简称Rho(x1,x2)为Rho, 变形得(E(X1*X2)-u1*u2)=Rho*s1*s2
代入(5)式的第三项得:Vp= w1^2*s1^2 + w2^2*s2^2 + 2*w1*w2*(Rho*s1*s2)
即得你要证明的公式。