Bienaymé's identity

In probability theory, the general form of Bienaymé's identity states that
 * $$\operatorname{Var}\left( \sum_{i=1}^n X_i \right)=\sum_{i=1}^n \operatorname{Var}(X_i)+2\sum_{i,j=1 \atop i < j}^n \operatorname{Cov}(X_i,X_j)=\sum_{i,j=1}^n\operatorname{Cov}(X_i,X_j)$$.

This can be simplified if $$X_1, \ldots, X_n$$ are pairwise independent or just uncorrelated, integrable random variables, each with finite second moment. This simplification gives:


 * $$\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{k=1}^n \operatorname{Var}(X_k)$$.

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.