Bienaymé's identity

In probability theory, the general^[1] form of Bienaymé's identity, named for Irénée-Jules Bienaymé, states that

$${\displaystyle {\begin{aligned}\operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)&=\sum _{i=1}^{n}\operatorname {Var} (X_{i})+2\sum _{i,j=1 \atop i<j}^{n}\operatorname {Cov} (X_{i},X_{j})\\&=\sum _{i,j=1}^{n}\operatorname {Cov} (X_{i},X_{j}).\end{aligned}}}$$

This can be simplified if ${\displaystyle X_{1},\ldots ,X_{n}}$ are pairwise independent or just uncorrelated, integrable random variables, each with finite second moment.^[2] This simplification gives:

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

The above expression is sometimes referred to as Bienaymé's formula. Bienaymé's identity may be used in proving certain variants of the law of large numbers.^[3]