## 3.7 Expected Values

SW 2.2

The **expected value** of some random variable \(X\) is its (population) mean and is written as \(\mathbb{E}[X]\). [I tend to write \(\mathbb{E}[X]\) for the expected value, but you might also see notation like \(\mu\) or \(\mu_X\) for the expected value.]

The expected value of a random variable is a *feature* of its distribution. In other words, if you know the distribution of a random variable, then you also know its mean.

The expected value is a measure of **central tendency** (alternative measures of central tendency are the **median** and **mode**).

Expected values are a main concept in the course (and in statistics/econometrics more generally). I think there are two main reasons for this:

Unlike a cdf, pdf, or pmf, the expected value is a single number. This means that it is easy to report. And, if you only knew one feature (at least a feature that that only involves a single number) of the distribution of some random variable, probably the feature that would be most useful to know would be the mean of the random variable.

Besides that, there are some computational reasons (we will see these later) that the mean can be easier to estimate than, say, the median of a random variable

If \(X\) is a discrete random variable, then the expected value is defined as

\[ \mathbb{E}[X] = \sum_{x \in \mathcal{X}} x f_X(x) \]

If \(X\) is a continuous random variable, then the expected value is defined as

\[ \mathbb{E}[X] = \int_{\mathcal{X}} x f_X(x) \, dx \] Either way, you can think of these as a weighted average of all possible realizations of the random variable \(X\) where the weights are given by the probability of \(X\) taking that particular value. This may be more clear with an example…

**Example 3.10 **Suppose that \(X\) is the outcome from a roll of a die. Then, its expected value is given by

\[ \begin{aligned} \mathbb{E}[X] &= \sum_{x=1}^6 x f_X(x) \\ &= 1\left(\frac{1}{6}\right) + 2\left(\frac{1}{6}\right) + \cdots + 6\left(\frac{1}{6}\right) \\ &= 3.5 \end{aligned} \]

Side-Comment: When we start to consider more realistic/interesting applications, we typically won’t know (or be able to easily figure out) \(\mathbb{E}[X]\). Instead, we’ll try to estimate it using available data. We’ll carefully distinguish between

population quantitieslike \(\mathbb{E}[X]\) andsample quantitieslike an estimate of \(\mathbb{E}[X]\) soon.