Conditional expectation

Conditional expectation

The conditional expectation of a discrete random variable $X$ given that the random variable $Y$ equals $y_i$ is given by: $$\mathbb{E}(X|Y=y_i) = \sum_{j=0}^\infty x_j P(X=x_j | Y=y_i )$$ Conditional expectation values are useful because we can use them to calculate absolute expectation values using the conditional expectation theorem: $$\mathbb{E}(X) = \sum_{i=0}^\infty \mathbb{E}(X|Y=y_i) P(Y=y_i)$$ Another useful result that is derived using similar logic involves the fact that the expectation $\mathbb{E}(XY)$ can be calculated as: $$\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$$ If the random variables $X$ and $Y$ are independent. This equation does not hold in general when the variables are not independent.

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