Wald's Equation

Let , ...,  be a sequence of independent observations of a random variable , and let the number of observations  itself be chosen at random. Then Wald's equation states that the expectation value of the sum  is equal to the expectation value of  times the expectation value of ,

(Wald 1945, Blackwell 1946, Wolfowitz 1947).

REFERENCES:

Blackwell, D. "On an Equation of Wald." Ann. Math. Stat. 17, 84-87, 1946.

Wald, A. "Sequential Tests of Statistical Hypotheses." Ann. Math. Stat. 16, 117-186, 1945.

Wolfowitz, J. "The Efficiency of Sequential Estimates and Wald's Equation for Sequential Processes." Ann. Math. Stat. 18, 215-230, 1947.