In Exercises 1 and 2, determine whether the random variable x is discrete or continuous. Explain.


Let x represent the grade on an exam worth a total of 100 points.

Verified step by step guidance

1. Understand the definitions: A discrete random variable takes on a countable number of distinct values, while a continuous random variable can take on any value within a given range, including fractions and decimals.
2. Identify the nature of the variable x: The grade on an exam worth a total of 100 points is typically expressed as a whole number (e.g., 85, 90, etc.), which suggests it is countable.
3. Consider whether fractional values are possible: If grades are rounded to whole numbers, x is discrete. However, if fractional grades (e.g., 85.5) are allowed, x could be continuous.
4. Determine the context: In most grading systems, grades are reported as whole numbers, making x a discrete random variable.
5. Explain the conclusion: Since the grade is typically a countable value and does not take on an infinite range of possibilities within an interval, x is classified as a discrete random variable.

Key Concepts

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. Examples include the number of students in a class or the outcome of rolling a die. In the context of grades, if we consider only whole number scores (0, 1, 2, ..., 100), the variable is discrete because it cannot take on values like 75.5.

Continuous Random Variable

A continuous random variable can take on an infinite number of values within a given range. This means it can represent measurements that can be infinitely divided, such as height or weight. If grades were measured with decimals (e.g., 75.5), then the variable would be considered continuous, as it could take any value within the range of possible scores.

Random Variable

A random variable is a numerical outcome 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 for analysis, such as probability distributions.