Final answer:
A random variable x is discrete if its values are countable and continuous if its values are measured and uncountable. Discrete random variables have probability distributions whose probabilities sum to one. Continuous random variables have probabilities calculated over intervals, not single points.
Step-by-step explanation:
To determine whether the random variable x is discrete or continuous, we need to consider how the variable's values are obtained. If the values can be counted, such as the number of books in a backpack or the number of miles driven, then the random variable is discrete. On the other hand, if the values are measured and can take on any value within a range, such as the weight of a book or the temperature on a particular day, then the random variable is continuous.
A discrete random variable has countable values and two main characteristics: (1) each probability is between zero and one, and (2) the sum of the probabilities equals one. An example of this is when x takes on the values of 0, 1, 2, 3, 4, 5, forming a discrete probability distribution.
For a continuous random variable, probabilities are always calculated over a range of values rather than for a single value because the probability for any single, exact value is zero. Therefore, we might express the probability that x lies within a certain interval, such as the probability that the height of a student is between 5'2" and 6'1".