Wald's Identity states that the expected value of the sum of a random number of independent, identically distributed random variables equals the expected value of one such variable multiplied by the expected value of the number of terms summed, provided that random number is a stopping time not depending on the future terms. Named for Abraham Wald, it is a foundational tool of sequential analysis used to compute expectations in random-sum settings, such as the expected total of a randomly stopped process, without summing the full distribution directly.
Facts
StatementThe expected value of a sum of a random number of independent, identically distributed finite-mean random variables equals the expected number of terms times the common expectation, when the number of terms is independent of the summands. 1 Sources
1. Wald's equation, Wikipedia
Lead, second sentence
In its simplest form, it relates the expectation of a sum of randomly many finite-mean, independent and identically distributed random variables to the expected number of terms in the sum and the random variables' common expectation under the condition that the number of terms in the sum is independent of the summands.
Lead, publication note
Wald published work on this topic in September 1944, as indicated by his paper "On cumulative sums of random variables" in The Annals of Mathematical Statistics.
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