Bienaymé's identity

In probability theory, the general[1] form of Bienaymé's identity states that

${\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{i=1}^{n}\operatorname {Var} (X_{i})+\sum _{i,j=1 \atop i\neq j}^{n}\operatorname {Cov} (X_{i},X_{j})=\sum _{i,j=1}^{n}\operatorname {Cov} (X_{i},X_{j})}$,

This can be simplified if ${\displaystyle X_{1},\ldots ,X_{n}}$ are pairwise independent integrable random variables with finite second moments[2]

${\displaystyle \operatorname {Var} (\sum _{i=1}^{n}X_{i})=\sum _{k=1}^{n}\operatorname {Var} (X_{k})}$

Bienaymé's identity may be used in proving certain variants of the law of large numbers.[3]