## 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 quantities like $$\mathbb{E}[X]$$ and sample quantities like an estimate of $$\mathbb{E}[X]$$ soon.