Law of total variance
In
probability theory, the
law of total variance states that if
X and
Y are
random variables on the same
probability space, and the
variance of
X is finite, then
(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). Similar comments apply to the conditional variance.)
The nomenclature used here parallels the phrase law of total probability. See also law of total expectation.
A similar law for the third central moment μ3 says
Generalizations for higher moments than the third are messy; for higher
cumulants on the other hand, a simple and elegant form exists.