Probability

Law of total expectation

Averages in two stages: within each scenario first, then across scenarios.

E[X]=E[ E[X∣Y] ]\mathbb{E}[X] = \mathbb{E}\big[\,\mathbb{E}[X \mid Y]\,\big]

Where

YY
Whatever, once known, makes XX easy — usually the first step or a hidden parameter.

Assumptions

  • Holds for any YY; the skill is choosing one that simplifies the inner expectation.

Where this is taught