Question

概率论方差题 D(X/3+Y/2)怎么算？

不独立。请问然后呢？就是不清楚中间是分数的，怎么应用公式？

Answer 1

D(X/3+Y/2)怎么算？
如果X,Y相互独立，那么
D(X/3 + Y/2) = D(X)/9 + D(Y)/4；
否则要用到X,Y的协方差：
令：Z=X/3+Y/2
D(Z)=D(X/3+Y/2)=E[(X-EX)²/9 + (Y-EY)²/4 + 2(X-EX)(Y-EY)/(2*3)]
= E[(X-EX)²/9] + E[(Y-EY)²/4] + E[(X-EX)(Y-EY)/3]
= D(X)/9 + D(Y)/4 + COV(X,Y)/3
其中：COV(X,Y) = E[(X-EX)(Y-EY)] 为X,Y的协方差。
如果x，y独立，那么cov（x，y）＝0.

Answer 2

D(X/3+Y/2)
=D(X/3)+D(Y/2)+2Cov(X/3,Y/2)
=D(X)/9+D(Y)/4+Cov(X,Y)/3