The proposition in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem,
among other names, states that if
is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
whose expected value
is defined, and
is any random variable on the same
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, then
:
i.e., the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the
conditional expected value
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
of
given
is the same as the expected value of
.
One special case states that if
is a finite or
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of the
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
, then
:
Note: The
conditional expected value
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
E(''X'' , ''Z'') is a random variable whose value depend on the value of ''Z''. Note that the conditional expected value of ''X'' given the ''event'' ''Z'' = ''z'' is a function of ''z''. If we write E(''X'' , ''Z'' = ''z'') = ''g''(''z'') then the random variable E(''X'' , ''Z'') is ''g''(''Z''). Similar comments apply to the conditional covariance.
Example
Suppose that only two factories supply
light bulb
An electric light, lamp, or light bulb is an electrical component that produces light. It is the most common form of artificial lighting. Lamps usually have a base made of ceramic, metal, glass, or plastic, which secures the lamp in the soc ...
s to the market. Factory
's bulbs work for an average of 5000 hours, whereas factory
's bulbs work for an average of 4000 hours. It is known that factory
supplies 60% of the total bulbs available. What is the expected length of time that a purchased bulb will work for?
Applying the law of total expectation, we have:
:
where
*
is the expected life of the bulb;
*
is the probability that the purchased bulb was manufactured by factory
;
*
is the probability that the purchased bulb was manufactured by factory
;
*
is the expected lifetime of a bulb manufactured by
;
*
is the expected lifetime of a bulb manufactured by
.
Thus each purchased light bulb has an expected lifetime of 4600 hours.
Proof in the finite and countable cases
Let the random variables
and
, defined on the same probability space, assume a finite or countably infinite set of finite values. Assume that