Bienaymé's identity

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

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

This can be simplified if ${\displaystyle X_{1},\ldots ,X_{n}}$ 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]

Estimated variance of the cumulative sum of iid normally distributed random variables (which could represent a gaussian random walk approximating a Wiener process). The sample variance is computed over 300 realizations of the corresponding random process.

See also

• Variance
• Propagation of error
• Markov chain central limit theorem
• Panjer recursion
• Inverse-variance weighting
• Donsker's theorem
• Paired difference test

References

1. Klenke, Achim (2013). Wahrscheinlichkeitstheorie. p. 106. doi:10.1007/978-3-642-36018-3.
2. Loève, Michel (1977). Probability Theory I. Springer. p. 246. ISBN 3-540-90210-4.
3. Itô, Kiyosi (1984). Introduction to Probability Theory. Cambridge University Press. p. 37. ISBN 0 521 26960 1.