> [!EXAMPLE] Example: Some simple expected values ^simple-ev > Recall the formula for the expected value of integer-valued random variables: > $ > \mathbb{E}(X) = \sum_{x} x \mathbb{P(X = x)} > $ > For each of the following random variables, compute their expected value. ###### Example 2.1. Suppose that $X$ has the following distribution | $x$ | -1 | 0 | 1 | | ------------------- | ----- | ----- | ----- | | $\mathbb{P}(X = x)$ | $1/6$ | $1/3$ | $1/2$ | Then $ \mathbb{E}(X) = (-1) (1/6) + (0) (1/3) + (1) (1/2) = \frac{1}{3} $ ###### Example 2.2. Suppose that $X$ has the following distribution | $x$ | -1 | 0 | 1 | | ------------------- | ----- | ----- | ----- | | $\mathbb{P}(X = x)$ | $1/4$ | $1/2$ | $1/4$ | Then $ \mathbb{E}(X) = (-1) (1/4) + (0) (1/2) + (1) (1/4) = 0 $ This distribution is symmetric about the value