- 概率证明题P73，22证明：当a<=X<=b时，有a
- 证明：当a<=X<=b时，有a<=EX<=b,0<=DX<=(b-a)^2/4
感谢！
- 使用这个性质:X≤Y==>E(X)≤E(Y).
1.
a≤X≤b==>a=E(a)≤E(X)≤E(b)=b.
2.
a≤X≤b==>[X-(a+b)/2]^2≤[(b-a)/2]^2
==>
E[X-(a+b)/2]^2≤E[(b-a)/2]^2=[(b-a)/2]^2
而
E[X-(a+b)/2]^2=
=E[X-E(X)+E(X)-(a+b)/2]^2=
=E[X-E(X)]^2+2E{[X-E(X),E(X)-(a+b)/2]}+E[E(X)-(a+b)/2]^2=DX+2[E(X)-(a+b)/2]E[X-E(X)]+[E(X)-(a+b)/2]^2
=DX+[E(X)-(a+b)/2]^2
==>
DX+[E(X)-(a+b)/2]^2≤(b-a)^2/4
==>
DX≤(b-a)^2/4