Final answer:
The domain of a function is all its possible input values, and the range is all possible outputs. For example, in the function f(x) = x^2, the domain is all real numbers, and the range is all real numbers ≥ 0. Random variables have specific domains, some of which are non-numerical, and all probabilities in a discrete probability distribution must sum to one and be between 0 and 1.
Step-by-step explanation:
The domain of a function is the set of all possible input values it can accept, while the range is the set of all possible output values it can produce.
Example:
Consider the function f(x) = x^2, where x is a real number. The domain of this function is all real numbers, because you can square any real number. The range, however, is all real numbers greater than or equal to zero, because the square of a real number is always non-negative.
Random variables like X (student's major), Y (number of classes taken), and Z (amount of money spent on books) can have domains that are not numerical. For instance, the domain of X is the set of all majors offered at the university plus 'undeclared'. The domain of Y consists of non-negative integers representing the count of classes, while the domain of Z includes all non-negative monetary amounts.
Anomalous data, such as a negative amount spent on books (e.g., z = -7), is not a possible value for Z since we cannot spend a negative amount of money. This highlights the importance of defining a sensible domain for our variables. Finally, the two essential characteristics of a discrete probability distribution are that the sum of the probabilities must equal one and each probability must be between 0 and 1.