Law of total expectation

The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, the smoothing theorem, and Adam's Law among other names, states that if X is an integrable random variable (i.e., a random variable satisfying E( |X| ) < ∞) and Y is any random variable, not necessarily integrable, on the same probability space, then

i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X.

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. If we write E( X | Y = y) = g(y) then the random variable E( X | Y ) is just g(Y).

One special case states that if ${\displaystyle A_{1},A_{2},\ldots ,A_{n}}$ is a partition of the whole outcome space, i.e. these events are mutually exclusive and exhaustive, then

Suppose that two factories supply light bulbs to the market. Factory X's bulbs work for an average of 5000 hours, whereas factory Y's bulbs work for an average of 4000 hours. It is known that factory X 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:

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