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Wald's equation
In probability theory, Wald's equation, Wald's identity or Wald's lemma is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities. 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.
The equation is named after the mathematician Abraham Wald. An identity for the second moment is given by the Blackwell–Girshick equation.
==Basic version==
Let be a sequence of real-valued, independent and identically distributed random variables and let be a nonnegative integer-value random variable that is independent of the sequence . Suppose that and the have finite expectations. Then
:
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