Lecture 3

Lecture

The Conditional Expectation of $X$ given $Y$

$$\mathbb{E}\left[X\mid Y=y_i\right]=\sum _i{x}_i\cdot P_{X\mid Y}(x_i\mid y_j)$$

The Law of Total Probability on Expectations is

$$\mathbb{E}\left[X\right]=\sum _j\mathbb{E}\left[X\mid Y=y_i\right]\cdot \mathbb{P}(Y=y_i)$$