Question

方差~急啊！！！在线等

随机变量x大于等于a，小于等于b，~证明Dx小于等于（b-a)^2/4

Answer 1

E[X-C]^2-D(X)=E[X-C]^2-E[X-E(X)]^2=[C-E(X)]^2>=0(C为任意数)
所以D(X)<=E[X-C]^2
令C=(a+b)/2
[X-(a+b)/2]^2<=(b-a)^2/4
所以D(X)<=E[X-C]^2<=E[(b-a)^2/4]=(b-a)^2/4