Conditional Expectation as a Random Variable
\[\mathbb{E}[X\lvert Y]=g(Y)\]
Law of Iterated Expectations
\[\mathbb{E}\left[\mathbb{E}[X\lvert Y]\right]=\mathbb{E}[X]\]
Law of Total Variance
\[\mathrm{var}(X)=\mathbb{E}[\mathrm{var}(X\lvert Y)]+\mathrm{var}\!\left(\mathbb{E}[X\lvert Y]\right)\]
Sum of a Random Number of Independent Random Variables
\[\mathrm{var}(Y)=\mathbb{E}[N]\,\mathrm{var}(X)+(\mathbb{E}[X])^2\,\mathrm{var}(N)\]