无偏估计的定义如下：
对随机变量
X
X
X的估计
X
ˉ
\bar{X}
Xˉ,如果
E
[
X
ˉ
]
=
E
[
X
]
E[\bar{X}] = E[X]
E[Xˉ]=E[X],则称
X
ˉ
\bar{X}
Xˉ是
X
X
X的无偏估计。
根据方差的定义展开得：
E
(
∑
i
=
1
n
(
X
i
−
X
ˉ
)
2
)
=
∑
i
=
1
n
E
(
X
i
2
)
−
n
E
(
X
ˉ
2
)
E(\sum_{i=1}^n(X_i-\bar{X})^2)= \sum_{i=1}^nE(X_i^2)-nE(\bar{X}^2)
E(i=1∑n(Xi−Xˉ)2)=i=1∑nE(Xi2)−nE(Xˉ2)
又根据
V
a
r
(
X
)
=
E
(
X
2
)
−
(
E
(
X
)
)
2
Var(X)=E(X^2)-(E(X))^2
Var(X)=E(X2)−(E(X))2有：
E
(
X
2
)
=
V
a
r
(
X
)
+
(
E
(
X
)
)
2
=
σ
2
+
μ
2
E(X^2)=Var(X)+(E(X))^2=\sigma^2+\mu^2
E(X2)=Var(X)+(E(X))2=σ2+μ2
E
(
X
ˉ
2
)
=
V
a
r
(
X
ˉ
)
+
(
E
(
X
ˉ
)
)
2
=
1
n
σ
2
+
μ
2
E(\bar{X}^2)=Var(\bar{X})+(E(\bar{X}))^2=\frac{1}{n}\sigma^2+\mu^2
E(Xˉ2)=Var(Xˉ)+(E(Xˉ))2=n1σ2+μ2
所以，
E
(
∑
i
=
1
n
(
X
i
−
X
ˉ
)
2
)
=
(
n
−
1
)
σ
2
E(\sum_{i=1}^n(X_i-\bar{X})^2)=(n-1)\sigma^2
E(i=1∑n(Xi−Xˉ)2)=(n−1)σ2
即
σ
2
=
1
n
−
1
E
(
∑
i
=
1
n
(
X
i
−
X
ˉ
)
2
)
\sigma^2=\frac{1}{n-1}E(\sum_{i=1}^n(X_i-\bar{X})^2)
σ2=n−11E(i=1∑n(Xi−Xˉ)2)
参考：/SoftPoeter/article/details/78273117