Final answer:
A random variable is discrete if its values are countable and obtained by counting, and continuous if its values result from measuring and are uncountable. Discrete variables have probabilities that sum up to one, whereas continuous variables are defined by ranges, since the probability of any single exact value is zero.
Step-by-step explanation:
To determine whether a random variable is discrete or continuous, you need to look at how its values are defined. A discrete random variable has countable values which are obtained by counting. For example, if X represents the number of books in a backpack, X is a discrete variable because you can count the number of books.
Conversely, a continuous random variable has uncountable values that result from measurement. For instance, if X signifies the weight of a book, X is continuous since weight is measured and can have an infinite number of values within a range.
Discrete probability distribution functions are characterized by two factors:
- Each probability is between zero and one, inclusive.
- The sum of the probabilities is one.
With continuous random variables, we never seek the probability of a specific value but rather the probability of falling within a range, since for any single value, P(x = c) is zero.