Final answer:
The domains of discrete random variables X, Y, and Z are sets of possible outcomes for a student's major, number of classes taken, and money spent on books, respectively. They're 'random' because their exact values are determined by unpredictable student choices. Value -7 is not valid for Z, and a discrete probability distribution's values must sum to one.
Step-by-step explanation:
Understanding the Domain of Discrete Random Variables
The concept of a domain is fundamental in understanding how discrete random variables function in probability and statistics. If we consider discrete random variables such as X, Y, and Z, which represent a student's major, the number of classes taken, and the amount of money spent on books, respectively, we can define their domains as follows:
- The domain of X consists of all the majors offered at a university, which might include English, Mathematics, etc.
- For Y, the domain is the set of whole numbers starting from zero, representing the possible number of classes a student might take.
- The domain of Z includes any monetary value starting from zero, reflecting the possible expenditures on books.
These variables are considered random because their outcomes are determined by the random nature of student choices and expenditures and can only be known after the data is collected. A special case where z equals -7 would not be a valid value for Z since you cannot spend a negative amount of money on books.
The two essential characteristics of a discrete probability distribution are the probability of each outcome and the requirement that the sum of all probabilities equals one. An example of a discrete probability distribution could be the recorded number of books checked out from a university library, where each count of books has a corresponding probability of occurring.