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Chapter 2: Problem 1

Decide whether a discrete or continuous random variable is the best model for each of the following variables: a. The number of cracks exceeding one-half inch in 10 miles of an interstate highway. b. The weight of an injection-molded plastic part. c. The number of molecules in a sample of gas. d. The concentration of output from a reactor. e. The current in an electronic circuit.

Short Answer

Expert verified
a) Discrete, b) Continuous, c) Discrete, d) Continuous, e) Continuous.

Step by step solution

01

Analyze the Definition of Discrete and Continuous Variables

A discrete random variable is one that can take on a countable number of values. Often these values are integers. Examples include the number of students in a class or the number of cars in a parking lot. On the other hand, a continuous random variable can take on any value within a given range, meaning it has an infinite number of possible values. Examples include temperature, time, and height.
02

Determine If Variable (a) is Discrete or Continuous

Variable (a) asks for the number of cracks exceeding one-half inch in 10 miles of a highway. The number of cracks is countable, so it is a discrete random variable.
03

Determine If Variable (b) is Discrete or Continuous

Variable (b) refers to the weight of an injection-molded plastic part. Since weight can take on any value within a range and is not countable in whole numbers, it is a continuous random variable.
04

Determine If Variable (c) is Discrete or Continuous

Variable (c) involves the number of molecules in a sample of gas. Molecules are countable, thus making it a discrete random variable.
05

Determine If Variable (d) is Discrete or Continuous

Variable (d) deals with the concentration of output from a reactor. Since concentration can vary continuously within a range of values, it is a continuous random variable.
06

Determine If Variable (e) is Discrete or Continuous

Variable (e) relates to the current in an electronic circuit. Current is measured in a continuous range, hence 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.

Discrete Random Variables
When talking about random variables, one of the first distinctions to make is between discrete and continuous random variables. Discrete random variables are a type of variable that can take on a specific number of distinct values. They are, by definition, countable. That means you can list all the possible values it can have. Think about scenarios where you have whole numbers or definitive counts. Here are some examples for clarity:
  • The number of students in a classroom.
  • The number of cars in a parking lot.
  • The example from our exercise: the number of cracks exceeding one-half inch in a 10-mile stretch of highway.
Whenever you come across a situation where you think to yourself, "Can I count this number of items or occurrences?" it's a clear sign you're dealing with a discrete random variable.
Continuous Random Variables
Continuous random variables, unlike discrete ones, can take on an infinite number of possible values. These values are often associated with measurements. Imagine measuring temperature in a room: it could be precisely 20掳C, but it might also be 20.1掳C or 19.95掳C. There's no end to how refined your measurement could be, allowing for infinite possibilities. This is a hallmark of continuous random variables. Some illustrative examples include:
  • Temperature readings at different times of the day.
  • Time taken to complete a task (as it can always be more precise).
  • In our exercise, the weight of an injection-molded plastic part and the concentration of output from a reactor are continuous because they can vary within a range.
Whenever you think about measurements that can be broken down into more precise values, you're thinking about continuous random variables.
Statistical Modeling
In statistical modeling, understanding the nature of your variables is vital. Such models are used to describe real-world phenomena, allowing us to predict, comprehend, and analyze data. Knowing whether a variable is discrete or continuous helps us choose the right statistical techniques and tools. For discrete random variables:
  • We might use probability mass functions (PMFs) or binomial distributions.
  • They are especially useful for count data and categorical outcomes.
For continuous random variables:
  • We might rely on probability density functions (PDFs) or normal distributions.
  • These are ideal for representing data that's measured and spans a continuum.
Proper variable identification streamlines our analysis, enables effective model selection, and enhances the accuracy of predictions.
Random Variable Identification
The process of random variable identification is crucial in statistical problem-solving. It involves distinguishing whether a variable is discrete or continuous based on its characteristics. Here are some steps and considerations:
  • Assess whether the values the variable can take are countable or infinite. If you can count them, it鈥檚 discrete. If they fall anywhere within a range, it鈥檚 continuous.
  • Consider the nature of the variable: is it about measuring something (e.g., length, weight) or counting (e.g., number of items)?
  • Review the context: for the exercise given, this involved considering the number of molecules in a gas sample (discrete) versus the current in an electronic circuit (continuous).
Precise identification ensures you apply the correct statistical tests and models. It's the backbone of effective data analysis and interpretation.

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Most popular questions from this chapter

A test of a printed circuit board uses a random test pattern with an array of 10 bits and each is equally likely to be 0 Or 1. Assume the bits are independent. a. What is the probability that all bits are \(1 \mathrm{~s}\) ? b. What is the probability that all bits are 0 s? c. What is the probability that exactly 5 bits are \(1 \mathrm{~s}\) and 5 bits are 0 s?

Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without distortion. Let \(A_{i}\) denote the event that the \(i\) th bit is distorted, \(i=1, \ldots, 4\) a. Describe the sample space for this experiment b. Are the \(A_{i}\) 's mutually exclusive? Describe the outcomes in each of the following events: c. \(A_{1}\) d. \(A_{1}^{\prime}\) e. \(A_{1} \cap A_{2} \cap A_{3} \cap A_{4}\) f. \(\left(A_{1} \cap A_{2}\right) \cup\left(A_{3} \cap A_{4}\right)\)

A recreational equipment supplier finds that among orders that include tents, \(40 \%\) also include sleeping mats. Only \(5 \%\) of orders that do not include tents do include sleeping mats. Also, \(20 \%\) of orders include tents. Determine the following probabilities: a. The order includes sleeping mats. b. The order includes a tent given it includes sleeping mats.

In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states and that is usually found in the following states: \(\begin{array}{ll}\text { Nickel Charge } & \text { Proportions Found }\end{array}\) $$ 0 $$ 0.17 $$ \begin{array}{rlr} +2 & 0.35 \\ +3 & 0.33 \\ +4 & 0.15 \end{array} $$ a. What is the probability that a cell has at least one of the positive nickel-charged options? b. What is the probability that a cell is not composed of a positive nickel charge greater than \(+3 ?\)

If \(P(A)=0.2, P(B)=0.2,\) and \(A\) and \(B\) are mutually exclusive, are they independent?
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