Question

Identify the following as discrete or continuous random variables: a. Increase in length of life attained by a cancer patient as a result of surgery b. Tensile breaking strength (in pounds per square inch) of 1 -inch-diameter steel cable c. Number of deer killed per year in a state wildlife preserve d. Number of overdue accounts in a department store at a particular time e. Your blood pressure

Short answer

Question: Identify if the following variables are discrete or continuous random variables: a. Increase in length of life attained by a cancer patient as a result of surgery b. Tensile breaking strength (in pounds per square inch) of 1-inch-diameter steel cable c. Number of deer killed per year in a state wildlife preserve d. Number of overdue accounts in a department store at a particular time e. Your blood pressure Answer: a. Continuous random variable b. Continuous random variable c. Discrete random variable d. Discrete random variable e. Continuous random variable

Step-by-step solution

Random variable: a. Increase in length of life attained by a cancer patient as a result of surgery

This variable can take on any value within a range, depending on how significantly the surgery impacts the patient's life. Since it can take on any value within a range, it is a continuous random variable.

Random variable: b. Tensile breaking strength (in pounds per square inch) of 1 -inch-diameter steel cable

This variable also represents a range of possible values, as the strength can vary from one cable to another. The tensile breaking strength can take any value within a range, on a continuous scale. Therefore, it is a continuous random variable.

Random variable: c. Number of deer killed per year in a state wildlife preserve

The number of deer killed per year is countable, and it cannot take on a fraction or decimal point. It can only take specific, whole number values. Thus, it is a discrete random variable.

Random variable: d. Number of overdue accounts in a department store at a particular time

The number of overdue accounts can be counted as whole numbers, as it cannot take on a fraction or decimal value. It must be a specific, countable number. This makes it a discrete random variable.

Random variable: e. Your blood pressure

Blood pressure can take on a range of possible values, even within fractions. It can change on a continuous scale depending on various factors like time, stress level, and more. Therefore, it is a continuous random variable.

Key concepts

Discrete Random Variables

When discussing discrete random variables, we refer to quantities that can take specific, separate values. These values are countable, typically integers. Discrete random variables are often found in scenarios where the possible outcomes can be listed. For example, imagine counting the number of students in a classroom. You can have 20, 21, or 22 students, but never 20.5 students. Similarly, in statistical exercises like the one above, the number of deer killed per year or overdue accounts in a department store are countable quantities.

Discrete random variables have interesting properties and features:

• They often arise in counting scenarios, where only whole numbers are valid.
• They do not allow for fractional or decimal results; a discrete variable cannot be 3.5 but can be 3 or 4.
• Probability functions are used to define the probability of discrete outcomes.

Understanding discrete random variables is vital in many areas of statistics and probability, as they help us work with and predict outcomes effectively.

Continuous Random Variables

Continuous random variables differ from their discrete counterparts, as they can take any value within a particular range. This means they can have an infinite number of possible outcomes within that range. Think of measuring someone's height, which could be any number of centimeters, even down to a fraction. In the context of the provided exercise, variables like tensile breaking strength and blood pressure are perfect examples of continuous random variables. These measurements can vary smoothly and continuously over a range.

Key characteristics of continuous random variables include:

• They are not restricted to separate, countable values; they can take on any value within a range.
• They are usually associated with measurements that can be highly precise.
• Probability density functions (PDFs) are employed to describe the probabilities of outcomes in continuous scenarios.

When dealing with continuous random variables, we often focus on determining the probability of the variable falling within a certain interval, rather than at an exact point, due to their continuous nature.

Probability Distributions

Probability distributions are mathematical functions that describe how probabilities are distributed over the values of a random variable. They are essential for understanding and analyzing random variables, whether discrete or continuous. For discrete random variables, we use probability mass functions (PMFs) to allocate probabilities to distinct outcomes. For continuous random variables, probability density functions (PDFs) model the likelihood of a variable falling within a particular range.

Important aspects of probability distributions include:

• They provide a complete description of the probabilities associated with every possible outcome or range of outcomes.
• For discrete variables, probabilities of each outcome must sum to 1.
• In continuous variables, the area under the probability density function over the entire range equals 1.

Understanding probability distributions enables us to make predictions, and to analyze statistical data. By modeling real-world phenomena accurately, distributions lay the foundation for inferential statistics and decision-making processes in various fields.