Intuitively obvious, Wald's theorem is a

technical result (with appropriately

boring proof) that says that a

stopping time behaves "correctly", and can't be used for

cheating when

gambling.

Let

*X*_{1},

*X*_{2}, ... be a sequence of

random variables with the same

distribution, and let

*T* be a

stopping time for them, which has

finite expectation.

Define
*S* = *X*_{1} + ... + *X*_{T}

(note that the number of

terms added to

*S* is itself a random variable; but this is well defined when

*T* is

almost surely finite, and we're even assuming finite expectation).
Then (the obvious for

expected values holds)

**E***S* = (

**E***T*)(

**E***X*_{1}).

When *T* is *not* a stopping time, it is easy to "cheat" and obtain very different expectation; see the nonexamples in my stopping time writeup, which all blatantly violate Wald's theorem.