Bienaymé's identity
In probability theory, the general form of Bienaymé's identity, named for Irénée-Jules Bienaymé, states that Var ( ∑ i = 1 n X i ) = ∑ i = 1 n Var ( X i ) + 2 ∑ i , j = 1 i < j n Cov ( X i , X j ) = ∑ i , j = 1 n Cov ( X i , X j ) . {\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 X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} are pairwise independent or just uncorrelated, integrable random variables, each with finite second moment.