Wald's equation

The formula of forest or Waldsche identity is in stochastics an equation with the help of the expectation value of sums of random variables can be computed with a random number of summands. It was demonstrated in 1944 in a forest work of the mathematician Abraham.

Formulation

It is with a sequence of independent, identically distributed, integrable random variables and one - valued random variable that is independent of the sequence. Then we have

Evidence

Because it is independent of the result by following condition to the value of:

So

By applying the expectation value of this equation one obtains finally

Generalization to stopping times

It is now a sequence of identically distributed integrable random variable which is adapted to a filtration, that is, for all is - measurable. If is independent of for all and an integrable stopping time with respect, so also applies the formula of the forest: