Bienaymé's identity

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.

In probability theory, the general form of Bienaymé's identity, named for Irénée-Jules Bienaymé, states that

$$\begin{array}{cc}\mathrm{Var}\left(\sum _{i=1}^n{X}_i\right)& =\sum _{i=1}^n\mathrm{Var}(X_i)+2\sum _{\genfrac{}{}{0ex}{}{i,j=1}{i<j}}^n\mathrm{Cov}(X_i,X_j)\\ & =\sum _{i,j=1}^n\mathrm{Cov}(X_i,X_j).\end{array}$$

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

$$\mathrm{Var}\left(\sum _{i=1}^n{X}_i\right)=\sum _{k=1}^n\mathrm{Var}(X_k).$$

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
• Inverse-variance weighting
• Donsker's theorem
• Paired difference test