Conditional Probability and Expectation Review

Conditional Probability:

P {A ∣ B} = \frac{P { A \cap B }}{P { B }}

for

P {B} > 0

.

Law of Total Probability:

P {A} = \sum_{i} P {A ∣ B_{i}} P {B_{i}}

for a partition

{B_{i}}

.

Conditional Expectation: Discrete:

E [X ∣ Y = y] = \sum x P {X = x ∣ Y = y}

. Continuous:

E [X ∣ Y = y] = \int x f_{X ∣ Y} (x ∣ y) d x

.

Law of Iterative Expectations:

E [X] = E [E [X ∣ Y]]

.