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.

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