13. Expected Value

Note

This is some text

Theorem 13.1 (Linearity of Expected Value)

Let \(X\) and \(Y\) be two independent random variables, then:

\[ \mathbb{E}[X + Y] = \mathbb{E}[X] + \mathbb{E}[Y] \]

Proof. By the definition of expected value:

\[\begin{split} \mathbb{E}[X + Y] & = \sum_{x, y \in X, Y} p(x, y) (x + y) \\ & = \sum_{x \in X} \sum_{y \in Y} p(x) p(y) (x + y) \\ & = \sum_{x \in X} \sum_{y \in Y} p(x) p(y) x + \sum_{x \in X} \sum_{y \in Y} p(x) p(y) y \\ & = \sum_{x \in X} p(x) x \sum_{y \in Y} p(y) + \sum_{y \in Y} p(y) y \sum_{x \in X} p(x) \\ & = \sum_{x \in X} p(x) x + \sum_{y \in Y} p(y) y \\ & = \mathbb{E}[X] + \mathbb{E}[Y] \end{split}\]