Bienaymé's identity explained

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

\operatorname{Var}\left( \sum ^{n} _{i =1} X _{i} \right) = \sum ^{n} _{i =1} \operatorname{Var}(X _{i)+2\sum} ^{n} ^{i,j =1 \atop i < j} \operatorname{Cov}(X _{i,X} _{j)=\sum} ^{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 ^{n} _{i =1} X _{i\right)} = \sum ^{n} _{k =1} \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.

See also

• Variance
• Propagation of error
• Markov chain central limit theorem
• Panjer recursion

References

1. Book: Klenke, Achim. 2013. Wahrscheinlichkeitstheorie. 106. 10.1007/978-3-642-36018-3.

^{[2] [3]}