Final answer:
The domain and range of functions relate to possible inputs and outputs for variables, shown through examples. Majors, class counts, and non-negative money spent are illustrative domains for random variables, emphasizing the specifics of the discrete probability distribution.
Step-by-step explanation:
Understanding the domain and range of a function is crucial in mathematics. The domain refers to the set of all possible input values (x-values) that a function can accept, while the range refers to the set of all possible output values (y-values) that a function can produce.
1. If X is a student's major, then the domain of X includes all the majors offered by the university, indicating a non-numerical set of data. This could be expressed as X = {English, Mathematics, ..., undeclared}.
2. If Y represents the number of classes taken in the previous semester, the domain of Y would be the set of all non-negative integers representing possible class counts. This is illustrated as Y = {0, 1, 2, ...}.
3. If Z signifies the amount of money spent on books in the previous semester, the domain of Z includes any non-negative monetary value. It can be any amount from zero upwards.
These examples show how X, Y, and Z are considered random variables because they can take on any value within their domain only after specific data is collected or an experiment is performed. Moreover, in a discrete probability distribution which Z exemplifies, a value like z = -7 would not be possible, since money spent cannot be negative. Therefore, this emphasizes the understanding that the range for this function includes only non-negative values.
The two essential characteristics of a discrete probability distribution involve having a finite or countably infinite list of values that the variable can take, along with associated probabilities that sum up to one.