Law of total expectation

The proposition in probability theory 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 whose expected value is defined, and is any random variable on the same probability space, then i.e., the expected value of the conditional expected value of given is the same as the expected value of . One special case states that if is a finite or countable partition of the sample space, then Note: The conditional expected value E(X | Y), with Y a random variable, is not a simple number; it is a random variable whose value depend on the value of Y. That is, the conditional expected value of X given the event Y = y is a number and it is a function of y. If we write g(y) for the value of E(X | Y = y) then the random variable E(X | Y) is g(Y). Suppose that only two factories supply light bulbs 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. When a joint probability density function is well defined and the expectations are integrable, we write for the general case A similar derivation works for discrete distributions using summation instead of integration. For the specific case of a partition, give each cell of the partition a unique label and let the random variable Y be the function of the sample space that assigns a cell's label to each point in that cell.