The conditional variance of a randomvariableY given another random variable X is The conditional variance tells us how much variance is left if we use to "predict" Y. Here, as usual, stands for the conditional expectation of Y given X, which we may recall, is a random variable itself. As a result, itself is a random variable.
Recall that variance is the expected squared deviation between a random variable and its expected value. The expected value can be thought of as a reasonable prediction of the outcomes of the random experiment. Thus, one interpretation of variance is that it gives the smallest possible expected squared prediction error. If we have the knowledge of another random variable that we can use to predict Y, we can potentially use this knowledge to reduce the expected squared error. As it turns out, the best prediction of Y given X is the conditional expectation. In particular, for any measurable, By selecting, the second, nonnegative term becomes zero, showing the claim. Here, the second equality used the law of total expectation. We also see that the expected conditional variance of Y given X shows up as the irreducible error of predictingY given only the knowledge of X.
When Xtakes on countable many values with positive probability, i.e., it is a discrete random variable, we can introduce, the conditional variance of Y given that X=x for any x from S as follows: where recall that is the conditional expectation of Z given that X=x, which is well-defined for. An alternative notation for is Note that here defines a constant for possible values of x, and in particular,, is not a random variable. The connection of this definition to is as follows: Let S be as above and define the function as. Then, almost surely.
The "conditional expectation of Y given X=x" can also be defined more generally using the conditional distribution of Y given X. In particular, letting be the conditional distribution of Y given X, i.e., , we can define This can, of course, be specialized to when Y is discrete itself, and also when the conditional density of Y given X=xwith respect to some underlying distribution exists.
Components of variance
The law of total variance says In words: the variance of Y is the sum of the expected conditional variance Y given X and the variance of the conditional expectation of Y given X. The first term captures the variation left after "using X to predict Y", while the second term captures the variation due to the mean of the prediction of Y due to the randomness of X.
