Discrete Variables and Continuous Variables In Exercises 13–18, determine whether the random variable x is discrete or continuous. Explain.


Let x represent the length of time it takes to complete an exam.

Verified step by step guidance

1. Understand the definitions: A discrete random variable takes on a countable number of distinct values (e.g., number of students in a class), while a continuous random variable can take on any value within a given range (e.g., height, weight, or time).
2. Identify the random variable in the problem: Here, the random variable x represents the length of time it takes to complete an exam.
3. Analyze the nature of the variable: Time is measured on a continuous scale and can take on any value within a range (e.g., 45.2 minutes, 60.75 minutes, etc.), including fractions of a second.
4. Determine whether the variable is discrete or continuous: Since time is not countable and can take on infinitely many values within a range, it is a continuous random variable.
5. Conclude and explain: The random variable x is continuous because it represents a measurement (time) that can take on any value within a range, rather than being limited to distinct, countable values.

Key Concepts

Discrete Variables

Discrete variables are those that can take on a countable number of distinct values. They often represent items that can be counted, such as the number of students in a class or the number of cars in a parking lot. In statistical analysis, discrete variables are typically represented using whole numbers.

Continuous Variables

Continuous variables, in contrast, can take on an infinite number of values within a given range. They represent measurements and can include fractions and decimals, such as height, weight, or time. For example, the length of time it takes to complete an exam can vary continuously, allowing for any value within a certain interval.

Random Variables

A random variable is a variable whose values are determined by the outcomes of a random phenomenon. It can be classified as either discrete or continuous based on the nature of its possible values. Understanding whether a random variable is discrete or continuous is crucial for selecting appropriate statistical methods and analyses.