## 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.