Bienaymé's identity

Bienaymé's identity states that the variance of a sum of independent random variables equals the sum of their variances.

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Variance: Variance measures the expected squared deviation of a random variable from its mean. The identity concerns how this measure behaves when random variables are added.
Random variable: A random variable assigns a numerical value to each outcome in a probability model. Bienaymé's identity applies to sums of random variables.
Irénée-Jules Bienaymé: Irénée-Jules Bienaymé was a French statistician who contributed to probability theory and statistics in the nineteenth century. His name is attached to the variance identity for independent sums.
Bernoulli random variable: A Bernoulli random variable takes value one with a specified probability and zero otherwise. Sums of independent Bernoulli variables are a standard setting for applying the identity.
Covariance: Covariance measures how two random variables vary together. The general variance-of-a-sum formula includes covariance terms that independence makes zero.
Linearity of expectation: Linearity of expectation states that the expectation of a sum equals the sum of expectations, without requiring independence. It underlies the expansion of the centered sum used to derive the identity.
Pierre-Simon Laplace: Pierre-Simon Laplace was a French mathematician whose work helped establish modern probability theory. His probability theory provided the setting for results about sums of random quantities.
Binomial distribution: The binomial distribution gives the number of successes in a fixed number of independent trials with the same success probability. Its variance follows by adding the variances of independent Bernoulli trials.
Independence (probability theory): Two random variables are independent when knowing one does not change the probability distribution of the other. Independence is the condition that eliminates cross terms in Bienaymé's identity.
Pairwise independence: A collection of random variables is pairwise independent when every pair is independent, even if the whole collection is not jointly independent. Pairwise independence suffices to make all covariance terms vanish in the identity.