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


Let x represent the populations of the 50 U.S. states.

Verified step by step guidance

1. Step 1: Understand the definitions of discrete and continuous variables. A discrete variable is one that can take on a countable number of distinct values, such as integers or specific categories. A continuous variable, on the other hand, can take on any value within a given range, including fractions and decimals.
2. Step 2: Analyze the random variable x in the problem. Here, x represents the populations of the 50 U.S. states.
3. Step 3: Consider the nature of population data. Population counts are typically whole numbers (e.g., 1,000,000 people) because you cannot have a fraction of a person in this context. This suggests that the variable is countable.
4. Step 4: Determine whether the variable is discrete or continuous. Since population values are countable and cannot take on an infinite range of values within a continuous interval, the variable x is discrete.
5. Step 5: Conclude that x is a discrete variable and explain why. The populations of the 50 U.S. states are represented by whole numbers, making x a discrete variable.

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 classroom or the populations of individual states. In this context, discrete variables can only assume whole numbers, making them suitable for scenarios where values cannot be subdivided.

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 or decimals, such as height, weight, or temperature. Continuous variables are typically associated with data that can be measured rather than counted, allowing for a more nuanced representation of information.

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, as it influences how data is interpreted and modeled.