Problem 183 - Applied Statistics and Probability for Engineers

Question

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.

Step-by-Step Solution

Answer

(a) Discrete, (b) Continuous, (c) Discrete, (d) Continuous, (e) Continuous.

Step 1: Understand Discrete and Continuous Variables

A discrete random variable has distinct, separate values, usually counts of items or events. A continuous random variable can take any value within a given range, and often involves measurements.

Step 2: Analyze Variable (a)

Variable (a) is 'the number of cracks exceeding one-half inch in 10 miles of an interstate highway'. Since this is a count of a distinct number of cracks, it is a discrete variable.

Step 3: Analyze Variable (b)

Variable (b) is 'the weight of an injection-molded plastic part'. Weight is a measurement that can take any value within a range and includes decimals, hence it is a continuous variable.

Step 4: Analyze Variable (c)

Variable (c) is 'the number of molecules in a sample of gas'. As it represents countable entities, even if very large, it is a discrete variable.

Step 5: Analyze Variable (d)

Variable (d) is 'the concentration of output from a reactor'. Concentration is a measurement that can take any real number within a specified range, making it a continuous variable.

Step 6: Analyze Variable (e)

Variable (e) is 'the current in an electronic circuit'. Current is a measurable value that can vary continuously, making it a continuous variable.

Key Concepts

Discrete Variables

Discrete variables are those that have separate, distinct values. Imagine counting something tangible like the cracks on a highway. Cracks can only form whole numbers, such as 0, 1, 2, etc. These are not fractional. Such counts are inherently finite and distinguish individual occurrences.
Examples include:

• The number of students in a classroom
• Number of cars parked in a garage
• Dice rolls outcome

Discrete variables are an essential part of probability models, especially when dealing with populations that involve counting items or events.

Continuous Variables

Continuous variables are measurements that can take any value within a range. Think of things you can measure, like the weight of a plastic part or the current flowing in a circuit.
In these cases, there aren't just whole numbers but a wide spectrum of possibilities, such as 1.25 pounds or 0.304 amps. Continuous variables are not countable in the discrete sense but can be any infinitesimally small point along a continuum.
Some fields where continuous variables are important include physics, chemistry, and engineering, where precise measurements are crucial for calculations.

Probability Models

Probability models are mathematical representations that depict real-world situations. They help us make sense of how random variables behave.
There are different models based on whether you have discrete or continuous variables. Discrete models, like the binomial or Poisson distributions, deal with countable outcomes. Continuous models, such as the normal distribution, handle uncountable results through calculus-based approaches.
Understanding what type of variable you are working with is critical to selecting the appropriate probability model, thus providing accurate predictions and insights.

Variable Analysis

Variable analysis is the process of determining the nature of a variable. Determining whether a variable is discrete or continuous can impact the statistical methods used.
For instance, analyzing a discrete variable like the number of molecules or highway cracks requires different tools than a continuous variable like weight or current. Analytic techniques focus on the variability, distribution, and behavior of these variables.
This knowledge facilitates constructing accurate models, helping scientists and statisticians to interpret data effectively and make informed decisions.