A recurring theme in probability theory is that sometimes plain language carries ambiguities that, when unpacked and laid bare, refer to multiple distinct mathematical notions. The reason agreed-upon mathematical definitions and precisely notation are so important is that they remove the possibility of disagreement arising from the interpretation of a question. Careful mathematical articulation replaces plain language ambiguity with clarity concerning the questions being asked. The word "average" is one of the worst offenders in this sense. >[!NOTE] Expected Value of an Integer-Valued Random Variable Let $X$ be a random variable that takes its values in the integers. We define the *expected value* of $X$ as follows: >$ >\mathbb{E}(X) \coloneqq \sum_{x\in\mathbb{Z}} x \mathbb{P}(X = x). >$ There are (at least) two helpful ways to think about this formula. - Physical: $\mathbb{E}(X)$ is the balance point of the probability distribution. - Economic: $\mathbb{E}(X)$ is the fair price of playing a probabilistic game. The first follows naturally from generalizing the idea behind the *sample mean* (or *arithmetic mean*) formula for a collection of data points. The second involves a shift of mindset and leads us to think about what people mean when they say "things will revert to the mean in the long run." Let's tackle the physical intuition first. ### Expected value as a balance point #### The average of a set of data points >[!note] Sample mean Suppose that we have a data set $\mathsf{x} = (x_1, x_2, \ldots, x_n)$. The sample mean $\bar{\mathsf{x}}$ is defined to be >$ >\bar{\mathsf{x}} \coloneqq \frac{1}{n}(x_1 + x_2 + \ldots + x_n). >$ #### Examples for averages of data sets ##### Grade Point Average When grade point averages are computed, each letter grade is assigned a numerical average and the GPA is the average of a students grade values. For the moment, let's ignore +'s and -'s. | A | B | C | D | F | | :-: | :-: | :-: | :-: | :-: | | 4 | 3 | 2 | 1 | 0 | *Problem.* Suppose that the grades for two students are summarized in the table below. | Student | Class 1 | Class 2 | Class 3 | Class 4 | Class 5 | | --------: | :-----: | :-----: | :-----: | :-----: | :-----: | | Student X | A | A | A | A | D | | Student Y | B | A | B | A | B | Make a numberline dot plot for each set of grade values. Based on the locations, estimate where you think the balance point would be for each. Calculate the GPA for each student. How was your balance point guess? ##### Movie Ratings IMDB rates movies on a scale from 0 to 10, but Rotten Tomatoes only allows users to rate movies on a Fresh vs Rotten scale. For the sake of the following challenges, suppose that users give ratings to movies using the integers $\{0, 1, \ldots, 9, 10\}$. If the rating is 7, 8, 9, or 10, it receives a `Fresh` rating for Rotten Tomatoes. Otherwise, it is given a `Rotten` rating. Suppose that there is a movie that registers 100% on the Rotten Tomatoes scale, but only 7.5 on IMDB.