Wald's equation

Wald's equation states that the expected sum of a random number of independent, identically distributed variables equals the expected count multiplied by their common mean, under suitable conditions.

Random sum: A sum whose number of terms is itself a random variable. Wald's equation gives a simple expectation formula for this kind of sum.
Expected value: The probability-weighted average of a random variable's possible values. Both the random count and the summed variables are evaluated through expectation.
Wald's sequential probability ratio test: A sequential hypothesis test that accumulates likelihood evidence until it crosses a decision boundary. Wald's equation can relate expected accumulated evidence to the test's expected sample size.
Optional stopping theorem: A collection of results giving conditions under which a stopped martingale retains its expectation. It concerns stopped processes broadly, while Wald's equation targets sums of i.i.d. increments.
Stopping time: A random time whose occurrence by each step can be determined from information available up to that step. The stopping-time condition is a common sufficient hypothesis for Wald's equation.
Independent and identically distributed random variables: A sequence of mutually independent random variables sharing one probability distribution. This standard sequence assumption combines the two key properties in Wald's equation.
Renewal theory: The study of processes that restart at random times after successive events. Wald's equation evaluates expected total increments over a random number of renewals.
Doob's optional stopping theorem: A theorem stating conditions under which the expectation of a martingale at a stopping time equals its initial expectation. It can derive forms of Wald's equation, but its martingale framework is more general.
Independence: A relation in which knowing the outcome of one random variable does not change the distribution of another. Independence between increments prevents the stopping rule from changing their common mean.
Partial sum: The sum of the first several terms of a sequence. A random sum can be written as a partial sum evaluated at a random index.

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Law of total expectationRelated: It is a specialized expectation identity closely connected to conditioning on the random count.