The Law of Total Variance Is Counterintuitive

The law of total variance states that for two random variables \(X\) and \(Y\), \[\mathrm{Var}(Y) = E[\mathrm{Var}(Y | X)] + \mathrm{Var}(E[Y | X])\] If \(Y\) is a mixture of distributions, this tells you that \(\mathrm{Var}(Y)\) is the mean variance of the components plus the variance of the means of the components (\(X\) in this case is a discrete variable indicating which component you…