Revisiting Conditioning

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

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