Law of Total Variance

The variance of \(X\) decomposes into the expected within-group variance and the variance of the group means:

\[\mathrm{var}(X)=\mathbb{E}[\mathrm{var}(X\lvert Y)]+\mathrm{var}\!\left(\mathbb{E}[X\lvert Y]\right)\]

Sum of a Random Number of Independent Random Variables

If \(Y=X_1+\ldots+X_N\) where the \(X_i\) are i.i.d. and \(N\) is itself a random variable independent of the \(X_i\),

\[\mathrm{var}(Y)=\mathbb{E}[N]\,\mathrm{var}(X)+(\mathbb{E}[X])^2\,\mathrm{var}(N)\]

Draft

This page is a partial draft — the full treatment is planned but not yet written.