/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 What is the difference between a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 2

What is the difference between a discrete random variable and a continuous random variable? Provide your own examples of each.

Short Answer

Expert verified
Discrete random variables have countable values; continuous random variables have values within a range. Examples: Number of cars (discrete), student heights (continuous).

Step by step solution

01

- Define a Discrete Random Variable

A discrete random variable is one that has countable values. This means the set of possible values can be listed out, like the outcomes of a roll of a die or the number of students in a class. Each individual outcome can be expressed as a distinct value.
02

- Example of a Discrete Random Variable

Consider the number of cars passing through a toll booth in an hour. This is a discrete random variable because you can count each car individually. For instance, you can have 10, 50, or 75 cars in a given hour, but you can't have 10.5 cars.
03

- Define a Continuous Random Variable

A continuous random variable, on the other hand, can take any value within a certain range. These values are not countable but instead measured. They can take on an infinite number of possible values within a selected range.
04

- Example of a Continuous Random Variable

An example would be the height of students in a classroom. This is a continuous random variable as it can take any value within a certain range (e.g., between 140 cm and 200 cm) and can include fractions of a unit (e.g., 165.2 cm or 170.5 cm).
05

- Summarize the Difference

In summary, the key difference is that discrete random variables have countable values with distinct gaps between them, while continuous random variables have values that can occupy any point within a range, forming a continuous spectrum.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Discrete Random Variable
A discrete random variable is a type of variable whose possible values can be counted. This means that you can list all possible outcomes. These outcomes are distinct and separate, with clear gaps between each value. For example, when you roll a fair six-sided die, the outcomes are 1, 2, 3, 4, 5, and 6. Each of these outcomes is specific and countable. Another common example is the number of students in a class. You might have 20, 30, or 35 students, but you cannot have 30.5 students in a typical setting.

A discrete random variable is particularly useful for situations in which you count occurrences, like the number of heads in a series of coin flips or the number of orders received in a day at a restaurant.
Continuous Random Variable
A continuous random variable, in contrast, can take any value within a given range. These values are typically measured rather than counted. As a continuous variable, it can include fractions and decimals, meaning there are an infinite number of possible values within the given range.

One example is measuring the height of students in a class. Heights could vary from, say, 140 cm to 200 cm. Within this range, a student鈥檚 height could be 150.2 cm or 178.5 cm. This variability and continuity make it so you cannot list all possible values since they form a continuum.

Continuous random variables are often used in natural and applied sciences to represent measurements like time, temperature, or distance.
Examples of Random Variables
Understanding and identifying random variables can help you in many real-world scenarios. Here are some practical examples to illustrate both discrete and continuous random variables:

Discrete Random Variable Examples:
  • Number of books on a shelf
  • Number of pets in a household
  • Number of questions answered correctly on a quiz
Each of these counts individual, separate occurrences.

Continuous Random Variable Examples:
  • Time taken to run a marathon (e.g., 4 hours, 3.5 hours)
  • Amount of milk in a carton (e.g., 1 liter, 1.25 liters)
  • Weight of a newborn baby (e.g., 3.2 kg, 3.35 kg)
These examples measure values within a range, with infinite possible outcomes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. Three cards are selected from a standard 52 -card deck with replacement. The number of kings selected is recorded.

According to flightstats.com, American Airlines flights from Dallas to Chicago are on time \(80 \%\) of the time. Suppose 15 flights are randomly selected, and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 10 flights are on time. (c) Find and interpret the probability that fewer than 10 flights are on time. (d) Find and interpret the probability that at least 10 flights are on time. (e) Find and interpret the probability that between 8 and 10 flights, inclusive, are on time.

BlackJack is a popular casino game in which a player is dealt two cards where the value of the card corresponds to the number on the card, face cards are worth ten, and aces are worth either one or eleven. The object is to get as close to 21 as possible without going over and have cards whose value exceeds that of the dealer. A blackjack is an ace and a ten in two cards. It pays 1.5 times the bet. The dealer plays last and must draw a card with sixteen and hold with seventeen or more. The following distribution shows the winnings and probability for a \(\$ 20\) bet. In cases where the dealer and player have the same value, there is a tie (called a "push"). Source: "Examining a Gambler's Claims: Probabilistic Fact-Checking and Don Johnson's Extraordinary Winning Streak" by W.J. Hurley, Jack Brimberg, and Richard Kohar. Chance Vol. 27.1,2014 $$ \begin{array}{cc} \text { Winnings } & \text { Probability } \\ \hline 0 & 0.0982 \\ \hline \$ 30 & 0.0483 \\ \hline \$ 20 & 0.389275 \\ \hline-\$ 20 & 0.464225 \end{array} $$ (a) Compute and interpret the expected value of the game from the player's point of view. (b) Suppose over the course of one hour, a player can expect to be dealt about 40 hands. How much should a player expect to win or lose over the course of three hours?

Historically, the probability that a passenger will miss a flight is 0.0995. Source: Passenger-Based Predictive Modeling of Airline No-show Rates by Richard D. Lawrence, Se June Hong, and Jacques Cherrier. Airlines do not like flights with empty seats, but it is also not desirable to have overbooked flights because passengers must be "bumped" from the flight. The Lockheed L49 Constellation has a seating capacity of 54 passengers. (a) If 56 tickets are sold, what is the probability 55 or 56 passengers show up for the flight resulting in an overbooked flight? (b) Suppose 60 tickets are sold, what is the probability a passenger will have to be "bumped"? (c) For a plane with seating capacity of 250 passengers, how many tickets may be sold to keep the probability of a passenger being "bumped" below \(1 \% ?\)

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=15, p=0.85, x=12 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.