Final answer:
A relation is a function if each input maps to one output. The domain of a variable includes all potential values it can take, while a random variable's value is determined by random processes. In probability, a discrete distribution's probabilities must sum to one and each one must be between 0 and 1.
Step-by-step explanation:
To determine if a relation is a function, we must check if every input (or domain value) maps to exactly one output (or range value). In this context, without the specific details about the relation, we can only explain the general concepts.
Understanding Domains and Ranges
The domain of a variable, such as a student's major (X), includes all possible values that the variable can take. For a student's major, the domain would be all the majors offered by an educational institution. The number of classes taken in the previous semester (Y) would have a domain of all non-negative integers, assuming one cannot take a negative number of classes. The amount of money spent on books (Z) should also have a domain of non-negative real numbers, as one cannot spend a negative amount of money.
Random Variables and Probability Distributions
Variables X, Y, and Z are considered random variables because their values are determined by the random outcomes of some process, such as a student choosing a major or purchasing books. For the amount of money spent on books, a negative value like z = -7 is not possible, as one cannot spend a negative amount of money on books, showing the importance of understanding domains.
Two essential characteristics of a discrete probability distribution are that the sum of the probabilities of all possible outcomes must equal one, and each individual probability must be between 0 and 1, inclusive.