The nomenclature used here parallels the phrase *law of total probability*. See also law of total variance.

(The conditional expected value E( *X* | *Y* ) is a random variable in its own right, whose value depends on the value of *Y*. Notice that the conditional expected value of *X* given the *event* *Y* = *y* is a function of *y* (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( *X* | *Y* = *y*) = *g*(*y*) then the random variable E( *X* | *Y* ) is just *g*(*Y*). )