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Wald's Equation

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Let X_1, ..., X_N be a sequence of independent observations of a random variable X, and let the number of observations N itself be chosen at random. Then Wald's equation states that the expectation value of the sum X_1+...+X_N is equal to the expectation value of X times the expectation value of N,

<X_1+...+X_N>=<X><N>

(Wald 1945, Blackwell 1946, Wolfowitz 1947).

See also

Expectation Value, Sum Distribution

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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.

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Wald's Equation

Cite this as:

Weisstein, Eric W. "Wald's Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WaldsEquation.html

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