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Theorem in probability theory
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.

Let (Xn)n∈ be a sequence of real valued, independent and identically distributed random variables and let N ≥ 0 be an integer valued random variable that is independent of the sequence (Xn)n∈. Suppose that N and the Xn have finite expectations. Then

Roll a six sided dice. Take the number on the die (call it N) and roll that number of six sided dice to get the numbers X1, . , XN, and add up their values. By Wald's equation, the resulting value on average is

Let N be an integrable, 0 valued random variable, which is independent of the integrable, real valued random variable Z with Define for all n ∈ Then assumptions , , , and with are satisfied, hence also and , and Wald's equation applies. If the distribution of Z is not symmetric, then does not hold. Note that, when Z is not almost surely equal to the zero random variable, then and cannot hold simultaneously for any filtration , because Z cannot be independent of itself as is impossible.

Let (Xn)n∈ be a sequence of independent, symmetric, and {–1, +1} valued random variables. For every n ∈ let be the σ algebra generated by X1, . , Xn and define when Xn is the first random variable taking the value +1. Note that , hence E[N] < ∞ by the ratio test. The assumptions , and , hence and with , , , and hold, hence also , and and Wald's equation applies. However, does not hold, because N is defined in terms of the sequence (Xn)n∈. Intuitively, one might expect to have E[SN] > 0 in this example, because the summation stops right after a one, thereby apparently creating a positive bias. However, Wald's equation shows that this intuition is misleading.

Consider a sequence (Xn)n∈ of i. i. d. (Independent and identically distributed random variables) random variables, taking each of the two values 0 and 1 with probability  (actually, only X1 is needed in the following). Define Then SN is identically equal to zero, hence , but and and therefore Wald's equation does not hold. Indeed, the assumptions , , and are satisfied, however, the equation in assumption holds for all n ∈ except for .

This proof uses only Lebesgue's monotone and dominated convergence theorems. We prove the statement as given above in three steps.

Wald's equation can be transferred to 'R'^(d) valued random variables (Xn)n∈ by applying the one dimensional version to every component. If (Xn)n∈ are Bochner integrable random variables taking values in a Banach space, then the general proof above can be adjusted accordingly.