Wald's equation

In probability theory, Wald's equation, Wald's identity[1] or Wald's lemma[2] 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.[3]

  1. ^ Janssen, Jacques; Manca, Raimondo (2006). "Renewal Theory". Applied Semi-Markov Processes. Springer. pp. 45–104. doi:10.1007/0-387-29548-8_2. ISBN 0-387-29547-X.
  2. ^ Thomas Bruss, F.; Robertson, J. B. (1991). "'Wald's Lemma' for Sums of Order Statistics of i.i.d. Random Variables". Advances in Applied Probability. 23 (3): 612–623. doi:10.2307/1427625. JSTOR 1427625. S2CID 120678340.
  3. ^ Blackwell, D.; Girshick, M. A. (1946). "On functions of sequences of independent chance vectors with applications to the problem of the 'random walk' in k dimensions". Ann. Math. Statist. 17 (3): 310–317. doi:10.1214/aoms/1177730943.