Final answer:
The domain of a relation defines the set of all possible inputs, while the range represents all possible outputs. If X represents student's major, Y the number of classes, and Z the amount spent on books, their domains would vary based on the context. X, Y, and Z are considered random variables because their values can vary and a value of z = -7 is not possible for Z.
Step-by-step explanation:
The domain and range of a relation help us understand the set of all possible inputs and outputs for that relation. The domain represents all possible input values while the range represents all possible output values.
- If X = student's major, then the domain of X would include all majors offered by an educational institution.
- If Y = the number of classes taken in the previous semester, the domain of Y would be a set of whole numbers representing the number of classes a student could possibly take, typically starting from 0.
- For Z = the amount of money spent on books in the previous semester, the domain of Z would be all non-negative real numbers since students can spend any amount greater than or equal to zero.
X, Y, and Z are considered random variables because they can take on different values depending on the situation or occurrence. A random variable is not fixed but varies from one instance to another.
An amount z = -7 for the money spent on books is not a possible value for Z because the domain for Z is non-negative real numbers; hence, negative amounts are outside the domain.
The two essential characteristics of a discrete probability distribution are that each possible value of the variable must be associated with a non-negative probability, and the sum of all probabilities must be 1.