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Chapter 11: Problem 6

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.

Short Answer

Expert verified
The random variable is discrete.

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

PUBLIC HEALTH As part of a campaign to combat a new strain of influenza, public health authorities are planning to inoculate 1 million people. It is estimated that the probability of an individual having a bad reaction to the vaccine is \(0.0005\). Suppose the number of people inoculated who have bad reactions to the vaccine is modeled by a random variable with a Poisson distribution. a. What is \(\lambda\) for the distribution? b. What is the probability that of the 1 million people inoculated, exactly five will have a bad reaction? c. What is the probability that of the 1 million people inoculated more than 10 will have a bad reaction?

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ECOLOGY The pH level of a liquid measures its acidity and is an important issue in studying the effects of acid rain. Suppose that a test is conducted under controlled conditions that allow the change in \(\mathrm{pH}\) in a particular lake resulting from acid rain to be recorded. Let \(X\) be a random variable that measures the \(\mathrm{pH}\) of a sample of water taken from the lake, and assume that \(X\) has the probability density function. \(f(x)= \begin{cases}0.75(x-4)(6-x) & \text { for } 4 \leq x \leq 6 \\ 0 & \text { otherwise }\end{cases}\) a. Find the probability that the \(\mathrm{pH}\) of a randomly selected sample will be at least \(5 .\) b. Find the expected \(\mathrm{pH}\) of a randomly selected sample.

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