Wald's equation

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Description:
In probability theory, Wald's equation (or Wald's identity) 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, 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.

Statement

Let be an infinite sequence of real-valued, finite-mean random variables and let N be a nonnegative integer-valued random variable. Assume that
  1. N has finite expectation,
  2. all have the same expectation,
  3. E<nowiki></nowiki>&nbsp;= E<nowiki></nowiki>&nbsp;P(N&nbsp;≥&nbsp;n) for every natural number n, and
  4. the infinite series satisfies
:<math>sum_^inftyoperatornamebigl<infty.</math>


Then the random sum
<math>S:=sum_^NX_n</math>
is integrable and
<math>operatorname=operatorname, operatorname.</math>


Discussion of assumptions

Clearly, assumptions (1) and (2) are needed to formulate Wald's equation. Assumption (3) controls the amount of dependence allowed between the sequence...
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