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Chapter 11: Problem 4

In each of Exercises 3 through 6 , determine whether the given random variable \(X\) is discrete or continuous. \(X\) measures the annual distance flown by a randomly selected airplane from a particular airline.

Short Answer

Expert verified
The random variable is continuous.

Step by step solution

01

Understanding the Random Variable

Review the random variable given in the problem: the annual distance flown by a randomly selected airplane from a particular airline. Determine what type of values this variable can take.
02

Defining Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. For example, the number of students in a class or the number of cars in a parking lot.
03

Defining Continuous Random Variable

A continuous random variable is one that can take on any value within a given range. These values are often measurements. Examples include the height of students in a class or the time it takes to run a race.
04

Analyzing the Given Variable

Consider whether the annual distance flown by an airplane is something that can be counted in distinct intervals (discrete) or measured and can take any value within a range (continuous).
05

Conclusion

Since the annual distance flown by an airplane can take any value within a range and is typically measured, it is a continuous random variable.

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Key Concepts

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

random variables
A random variable is a fundamental concept in statistics and probability. It represents a variable whose possible values are numerical outcomes of a random phenomenon. For example, consider rolling a die; the result, which can be any number from 1 to 6, represents a random variable.
Random variables allow statisticians to describe outcomes from random processes in a structured way. They can be further classified into two main types: discrete and continuous random variables.
Breaking down random variables makes analyzing and predicting complex random outcomes easier.
discrete random variable
A discrete random variable can take on a finite or countable number of distinct values. Each value is separate and distinct, which means there are gaps between possible values. For instance, the number of students in a classroom or the number of cars in a parking lot are examples of discrete random variables.
Key characteristics of discrete random variables include:
  • Countable values: You can list each possible value or count them easily.
  • No intermediate values: There are no possibilities between distinct values (no decimals or fractions).
  • Examples: Rolling a die (outputs 1, 2, 3, 4, 5, or 6), flipping a coin (outputs heads or tails).
Understanding discrete random variables is crucial for grasping how they differ from continuous random variables.
continuous random variable
A continuous random variable can take on any value within a given range. Unlike discrete random variables, the possible values for continuous random variables form an entire interval, meaning they can take on an infinite number of possible outcomes. For example, the height of students in a classroom or the time it takes to complete a race are continuous random variables.
Key characteristics of continuous random variables include:
  • Infinite possible values: They can take any value within a range with no gaps between values.
  • Measurement-based: Typically measured rather than counted, which often involves decimals and fractions.
  • Examples: Annual distance flown by an airplane, temperature readings, weights of objects.
Recognizing continuous random variables is essential for understanding variables that involve measurements that can vary within ranges.

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Most popular questions from this chapter

In Exercises 1 and 2, the outcomes and corresponding probability assignments for a discrete random variable \(X\) are listed. Draw the histogram for \(X\). Then find the expected value \(E(X)\), the variance \(\operatorname{Var}(X)\), and the standard deviation \(\sigma(X)\). $$ \begin{array}{l|c|c|c|c|c} \hline \text { Outcomes for } X & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Probability } & \frac{1}{9} & \frac{2}{9} & \frac{1}{3} & \frac{1}{9} & \frac{2}{9} \\ \hline \end{array} $$

INSURANCE POLICY An insurance company charges \(\$ 10,000\) for a policy insuring against a certain kind of accident and pays \(\$ 100,000\) if the accident occurs. Suppose it is estimated that the probability of the accident occurring is \(p=0.02\). Let \(X\) be the random variable that measures the insurance company's profit on each policy it sells. a. What is the probability distribution for \(X\) ? b. What is the company's expected profit per policy sold? c. What should the company charge per policy to double its expected profit per policy?

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.

Find \(b\) if \(P(Z \geq b)=0.73\), where \(Z\) is a random variable with a standard normal distribution \((\mu=0, \sigma=1)\).

QUALITY CONTROL A toy manufacturer makes hollow rubber balls. The thickness of the outer shell of such a ball is normally distributed with mean \(0.03\) millimeter and standard deviation \(0.0015\) millimeter. What is the probability that the outer shell of a randomly selected ball will be less than \(0.025\) millimeter thick?
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