概率论方差题 D(X⼀3+Y⼀2)怎么算?

2025-03-04 19:09:07
推荐回答(2个)
回答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.

回答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