Wald's equation

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Wald's equation relates the expectation of a random sum of i.i.d. random variables to the expectations of the number in the sum and the random variables' common expectation.

Let X1, X2, ..., XN be a sequence of (a random number) N i.i.d. random variables distributed identically to some random variable X, such that

  1. N > 0 is itself a random variable (integer-valued),
  2. the expectation of X, E(X) < ∞, and
  3. E(N) < ∞.

Then

\operatorname{E}\left(\sum_{i=1}^{N}X_i\right)=\operatorname{E}(N)\operatorname{E}(X).

In the general case, the random number N can be a stopping time for the stochastic process { Xi, i = 1, 2, ... }.

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This article incorporates material from Wald's equation on PlanetMath, which is licensed under the GFDL.


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