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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.

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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.

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Most popular questions from this chapter

An investment counselor calls with a hot stock tip. He believes that if the economy remains strong, the investment will result in a profit of \(\$ 50,000\). If the economy grows at a moderate pace, the investment will result in a profit of \(\$ 10,000 .\) However, if the economy goes into recession, the investment will result in a loss of \(\$ 50,000\). You contact an economist who believes there is a \(20 \%\) probability the economy will remain strong, a \(70 \%\) probability the economy will grow at a moderate pace, and a \(10 \%\) probability the economy will slip into recession. What is the expected profit from this investment?

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section \(6.1 ;\) (c) compute the mean and standard deviation, using the methods of this section; and \((d)\) draw a graph of the probability distribution and comment on its shape. $$ n=9, p=0.75 $$

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=12, p=0.35, x \leq 4 $$

The random variable \(X\) follows a Poisson process with the given mean. Assuming \(\mu=5,\) compute (a) \(P(6)\) (b) \(P(X<6)\) (c) \(P(X \geq 6)\) (d) \(P(2 \leq X \leq 4)\)

In a recent poll, the Gallup Organization found that \(45 \%\) of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor. (c) Find and interpret the probability that more than 16 of those surveyed feel the state of morals is poor. (d) Find and interpret the probability that 13 or 14 believe the state of morals is poor. (e) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values is poor in the United States? Why?
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