Question

连续型随机变量x的一切可能取值[a,b]内,其密度函数为f(x).求证：4D(X)

Answer 1

首先证明E(X-C)^2,当C=EX时最小,最小值为D(X)

E(X-C)^2=E[(X-EX)+(EX-C)]^2

=D(X)+2(EX-C)E(X-EX)+(EX-C)^2

=D(X)+(EX-C)^2

故当C=EX时,E(X-C)^2最小,最小值为D(X)

D(X)=E(X-EX)^2