Bienaymé's identity
Appearance
In probability theory, the general[1] form of Bienaymé's identity states that
- .
This can be simplified if are pairwise independent or just uncorrelated, integrable random variables, each with finite second moment.[2] This simplification gives:
- .
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]
![](http://up.wiki.x.io/wikipedia/commons/thumb/8/84/Estimated_and_expected_variance_of_cumulative_sum.png/220px-Estimated_and_expected_variance_of_cumulative_sum.png)
See also
[edit]References
[edit]- ^ Klenke, Achim (2013). Wahrscheinlichkeitstheorie. p. 106. doi:10.1007/978-3-642-36018-3.
- ^ Loève, Michel (1977). Probability Theory I. Springer. p. 246. ISBN 3-540-90210-4.
- ^ Itô, Kiyosi (1984). Introduction to Probability Theory. Cambridge University Press. p. 37. ISBN 0 521 26960 1.