Problem 6 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In each of Exercises 3 through 6 , determine whether the given random variable \(X\) is discrete or continuous. \(X\) counts the number of books in the library of a randomly selected professor at your school.

Step-by-Step Solution

Answer

The random variable is discrete.

Step 1: Define Discrete and Continuous Variables

A discrete variable is one that has specific, separate values. It can't take on all values within a given range. Typically, it is countable. In contrast, a continuous variable can take on any value within a range, typically measurable.

Step 2: Analyze the Random Variable

Consider the variable in question: it counts the number of books in the library of a randomly selected professor. Since we are counting the number of books, it involves specific, individual amounts.

Step 3: Determine the Variable Type

Because the number of books can be enumerated (1 book, 2 books, 3 books, etc.) and there cannot be fractions of a book that count, the variable can only take specific, separate values. Hence, it is a discrete random variable.

Key Concepts

random variable

A random variable is a numerical outcome of a random phenomenon. It assigns a number to each possible outcome of a random process. For example, when you roll a die, the face showing up is a random outcome, and we can assign numbers 1 to 6 to these outcomes.

There are two main types of random variables: discrete and continuous. The type depends on the values the variable can assume. Essentially, if the outcomes are countable, it's a discrete random variable. If they can take any value within a range, it's continuous.

discrete variable

A discrete variable can only take specific, separate values. These values are countable and not continuous. For instance, the number of students in a classroom can be 20, 21, 22, and so on, but not 20.5.

When dealing with discrete variables, we often use them to count distinct items or events. The key characteristic here is that you're counting, not measuring. Also, there's a clear separation between possible values.

continuous variable

A continuous variable, in contrast, can take any value within a given range. Unlike discrete variables, continuous variables are measurable and can have an infinite number of possible values within a certain interval. For example, height, weight, and time are continuous variables because they can take on virtually any value within a certain range.

Continuous variables are used when dealing with measurements. Think of them as numbers that can always be more precise if you use a more precise tool.

countable values

Countable values refer to the distinct and separate values that can be listed or enumerated. For discrete variables, these values are countable, meaning we can list them as 1, 2, 3, and so on.

In the context of the example problem, the number of books is a countable value. You can list 1 book, 2 books, 3 books, etc. Hence, when you deal with a variable that takes countable values, you are dealing with a discrete random variable.