Probability Adventures #16: Conditional expectation

Consider two random variables X and Y. Then the mean of X is a number. But if you knew the value of Y, that information could change the distribution of X. For instance, it could be that [X | Y] is uniform over the interval from 0 to Y. Then the mean of [X | Y] would be (0 + Y) / 2 = Y / 2. That is a function of the random variable Y, and in general, the mean of [X | Y] will always be a function of Y. It gets even better, if we then take the mean of that function of Y, the end result is the mean of the original X.

That idea, that the mean of the mean of X given Y is just the mean of X, is the *Fundamental Theorem of Probability*, and it can be used to derive many important results in probability.