## 4.2 Estimating \(\mathbb{E}[Y]\)

SW 2.5, 3.1

Let’s start with trying to estimate \(\mathbb{E}[Y]\) as this is probably the simplest, non-trivial thing that we can estimate.

A natural way to estimate population quantities is with their sample analogue. This is called the **analogy principle**. This is perhaps technical jargon, but it is the way you would immediately think to estimate \(\mathbb{E}[Y]\):

\[ \hat{\mathbb{E}}[Y] = \frac{1}{n} \sum_{i=1}^n Y_i = \bar{Y} \] In this course, we will typically put a “hat” on estimated quantities. The expression \(\displaystyle \frac{1}{n}\sum_{i=1}^n Y_i\) is just the average value of \(Y\) in our sample. Since we will calculate a ton of averages like this one over the course of the rest of the semester, it’s also convenient to give it a shorthand notation, which is what \(\bar{Y}\) means — it is just the sample average of \(Y\).

One thing that is important to be clear about at this point is that, in general, \(\mathbb{E}[Y] \neq \bar{Y}\). \(\mathbb{E}[Y]\) is a population quantity while \(\bar{Y}\) is a sample quantity. We will hope (and provide some related conditions/discussions below) that \(\bar{Y}\) would be close to \(\mathbb{E}[Y]\), but, in general, they will not be exactly the same.