Final answer:
The domain of a random variable consists of all possible values it can take, such as all majors for X, non-negative integers for Y, and non-negative real numbers for Z. Random variables represent the outcomes of statistical experiments and must satisfy certain conditions in a discrete probability distribution. A value of z = -7 is not possible for Z as it represents a negative amount of money spent.
Step-by-step explanation:
The domain of a random variable represents all the possible values that the variable can take. In the context of the given examples:
- The domain of X, which represents a student's major, would include all possible majors offered at the institution.
- The domain of Y, which signifies the number of classes taken in the previous semester, would be a set of non-negative integers since you cannot take a negative number of classes.
- The domain of Z, the amount of money spent on books in the previous semester, would be non-negative real numbers since you cannot spend a negative amount of money.
For the geometric experiment where you draw cards without replacement until you draw a red card, the domain of X would be the set {1, 2, 3, ..., 17} because there are 26 red cards in a deck; you must draw one within the first 17 draws to ensure that you get a red card, as you might draw all 26 black cards first.
Random variables like X, Y, and Z are quantities that can assume different values depending on the outcomes of a statistical experiment. They are considered random because the result is not deterministic until the experiment is performed.
A discrete probability distribution, including the one you mentioned, must satisfy two essential characteristics: the sum of all probabilities must equal 1, and each individual probability must be between 0 and 1, inclusive.
The finding of z = -7 is not a possible value for Z, as the amount of money spent cannot be negative.