Lecture
In probability theory, the statement known as the law of total expectation , law of iterated expectations , tower rule , Adam's law or smoothing theorem states that if 
is a random variable with a defined expectation 
, and 
is an arbitrary random variable on the same probability space.
that is, the expectation of the conditional expectation of given equals the expectation of .
In the special case where 
is a finite or countable partition of the sample space, then
Suppose that two factories supply light bulbs to the market. Light bulbs from factory work on average 5000 hours, while bulbs from factory work on average for 4000 hours. It is known that factory supplies 60% of the total number of bulbs available. What is the expected lifetime of a purchased light bulb?
Applying the law of total expectation we get:
where
is the lifetime of the bulb;
is the probability that the purchased bulb was made at factory X;
is the probability that the purchased bulb was made at factory Y;
is the expected lifetime of a bulb made at factory X;
is the expected lifetime of a bulb made at factory Y.Therefore, the expected lifetime of each purchased light bulb equals 4600 hours.
Let the random variables and be defined on the same probability space, taking a finite or countable set of finite values. Suppose that 
is defined, that is 
. If 
is a partition of the probability space Ω , then
If the series is finite, we can change the order of summation and the previous expression becomes
If, on the other hand, the series is infinite, its convergence cannot be conditional because of the assumption that 
The series converges absolutely if both, 
and 
are finite, and diverges to infinity if either or is infinite. In both cases, the order of summation can be changed without changing the sum.
Let 
be a probability space with σ-algebras 
defined on it. For a random variable on such a space, the smoothing theorem states that if is defined, that is , then
. 
(almost surely)
Proof . Since the conditional expectation is a Radon–Nikodym derivative, proving the smoothing theorem reduces to verifying the following two properties:
is 
-measurable
for all 
The first of these properties follows from the definition of conditional expectation. To prove the second,
hence the integral 
is defined (not equal to±∞
).
The second property holds because 
implies
Conclusion. In the special case where 
and )
, the smoothing theorem reduces to
where 
is the indicator function of the set 
.
If the partition 
is finite, then, by the linearity property, the preceding expression is written as
which is what needed to be shown.
If, on the other hand, the partition 
is infinite, then applying the dominated convergence theorem we can show
Indeed, for every 
,
Since every element of the setΩ belongs to a specific element of the partition , it is easy to verify that the sequence 
converges pointwise to X. By the assumption of the statement, 
. Applying the dominated convergence theorem gives the desired result.
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