The Generalized Laws of Total Variance and Total Covariance

The law of total variance states that the unconditional variance of a random variable Y  is the sum of (a) the variance of the conditional expectation of Y  given X and (b) the expectation of the conditional variance of Y  given the random variable X. We show that the total variance of Y  can be partitioned by using the relationship between Y  and one or more random variables \(X_{1},\ldots ,X_{k}\) , where \(k\geq 1\) . An application in multivariate analysis is given. Further, we generalize the total law of total covariance and show that the generalized law of total variance is a special case. Some examples are referenced.