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Chapter 4: Problem 80

Identify the following as discrete or continuous random variables: a. Total number of points scored in a football game b. Shelf life of a particular drug c. Height of the ocean's tide at a given location d. Length of a 2-year-old black bass e. Number of aircraft near-collisions in a year

Short Answer

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**Question:** Classify each of the following random variables as discrete or continuous: a. Total number of points scored in a football game b. Shelf life of a particular drug c. Height of the ocean's tide at a given location d. Length of a 2-year-old black bass e. Number of aircraft near-collisions in a year **Answer:** a. Discrete b. Continuous c. Continuous d. Continuous e. Discrete

Step by step solution

01

a. Total number of points scored in a football game

The number of points scored in a football game is countable as it consists of a series of specific, distinct values (integers). Therefore, this random variable is discrete.
02

b. Shelf life of a particular drug

The shelf life of a drug is a measurement of time, which can be expressed as a decimal value. Hence, it can take on any value within a specified range (e.g., between 0 and 10 years). This makes it a continuous random variable.
03

c. Height of the ocean's tide at a given location

The height of the ocean's tide is a measurement of length, which can be expressed as a decimal value. Like the shelf life of a drug, it can take on any value within a specified range (e.g., between 0 and 50 feet). Therefore, this random variable is continuous.
04

d. Length of a 2-year-old black bass

The length of a 2-year-old black bass is also a measurement that can be expressed as a decimal value. It can take on any value within a specified range (e.g., between 5 and 30 inches). This makes it a continuous random variable.
05

e. Number of aircraft near-collisions in a year

The number of near-collisions is countable, as it consists of a series of specific, distinct values (integers). Therefore, this random variable is discrete.

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Key Concepts

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

Discrete Variables
Discrete variables are those that can take on a finite or countable number of distinct values. These values are often whole numbers due to their nature of being countable. For instance, in a football game, counting the total number of points scored gives you discrete values like 0, 1, 2, 3, and so on. Such variables are measurable using integers. Characteristics of discrete variables include:
  • Distinct and separate values
  • Often represented by whole numbers
  • Can be counted but not measured precisely as fractions or decimals
In practical situations, you might encounter discrete variables in various forms:
  • Number of students in a classroom
  • Number of books on a shelf
  • Number of cars in a parking lot
By understanding the nature of discrete variables, you can easily identify and work with them in different statistical contexts.
Continuous Variables
Continuous variables, on the other hand, represent data that can take on any value within a given range. Because these variables can have an infinite number of potential values, they can be represented with decimals, allowing for very detailed and precise measurements. An example of this concept can be seen in the shelf life of a drug or the height of the ocean's tide. Both are examples of continuous random variables because they can assume any value:
  • A drug's shelf life could be measured precisely down to days or even hours.
  • The ocean's tide height could be measured to millimeters according to fluctuations.
Characteristics of continuous variables include:
  • An infinite range of possible values
  • Can be measured precisely and expressed in decimals or fractions
  • Sensitive to the level of measurement precision
These variables are common in fields requiring high precision, such as pharmaceuticals and environmental science, where detailed precision is crucial.
Probability Concepts
Probability concepts in statistics revolve around understanding how likely an event is to occur. In the context of random variables, probability helps us predict outcomes based on quantitative patterns. There are two primary types of probability distributions:
  • **Discrete probability distribution**: Used with discrete variables, where outcomes are countable and finite. Examples include the roll of a die or the number of students who pass a test.
  • **Continuous probability distribution**: Used with continuous variables where any value within a range is possible. Examples include the probability of someone's exact height or the precise amount of rain in a day.
Each type has its insights and applications:
  • Discrete distributions can be calculated using tools like probability mass functions.
  • Continuous distributions are described with probability density functions.
By grasping these fundamental probability concepts, you can better analyze and predict patterns and outcomes in data involving discrete and continuous random variables.

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Most popular questions from this chapter

An experiment can result in one of five equally likely simple events, \(E_{1}, E_{2}, \ldots, E_{5} .\) Events \(A, B,\) and \(C\) are defined as follows: $$A: E_{1}, E_{3} \quad P(A)=.4$$ $$B: E_{1}, E_{2}, E_{4}, E_{5} \quad P(B)=.8$$ $$C: E_{3}, E_{4} \quad P(C)=.4$$. Find the probabilities associated with these compound events by listing the simple events in each. a. \(A^{c}\) b. \(A \cap B\) c. \(B \cap C\) d. \(A \cup B\) e. \(B \mid C\) f. \(A \mid B\) g. \(A \cup B \cup C\) h. \((A \cap B)^{c}\)

Permutations Evaluate the following permutations. a. \(P_{3}^{5}\) b. \(P_{9}^{10}\) c. \(P_{6}^{6}\) d. \(P_{1}^{20}\)

A certain manufactured item is visually inspected by two different inspectors. When a defective item comes through the line, the probability that it gets by the first inspector is \(.1 .\) Of those that get past the first inspector, the second inspector will "miss" 5 out of \(10 .\) What fraction of the defective items get by both inspectors?

Five cards are selected from a 52 -card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the \(\mathrm{A}, \mathrm{K}, \mathrm{Q}, \mathrm{J},\) and \(10,\) all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush?

An experiment involves tossing a single die. These are some events: A: Observe a 2 \(B:\) Observe an even number \(C:\) Observe a number greater than 2 \(D:\) Observe both \(A\) and \(B\) \(E:\) Observe \(A\) or \(B\) or both \(F:\) Observe both \(A\) and \(C\) a. List the simple events in the sample space. b. List the simple events in each of the events \(A\) through \(F\) c. What probabilities should you assign to the simple events? d. Calculate the probabilities of the six events \(A\) through \(F\) by adding the appropriate simple-event probabilities.
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