Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The distance in miles a commuter travels to work

Step-by-Step Solution

Answer

The random variable $X$ has a range from 0 to 150 miles (\(X \in [0, 150]\)), and it is classified as a continuous random variable.

Step 1: Determine the range of values for the random variable X

The random variable X represents the distance (in miles) a commuter travels to work. From a practical point of view, the distance cannot be negative, so the lower limit is zero (0). There is no strict upper limit, but we could assume a reasonable maximum distance for a daily commute, for example 150 miles. Therefore, the range of values for X will be from 0 to 150 miles, including both endpoints. We can represent this range using the notation: \(X \in [0, 150]\).

Step 2: Classify the random variable X

As the random variable X represents the distance a commuter travels to work, it can be assumed that the distances might have decimal values, and not restricted to only integers. Moreover, there are infinite possible decimal values in between two integers. So the random variable X is neither finite discrete nor infinite discrete. Since distances can assume any value in the continuous range from 0 to 150 miles, the random variable X is continuous. Final answer: The random variable X has a range from 0 to 150 miles, and it is classified as a continuous random variable.

Key Concepts

Continuous Random Variable

A continuous random variable is one that can take on an infinite number of values within a given range. Unlike discrete variables that take specific values, continuous random variables can have any value within an interval.
In the given exercise, the distance in miles a commuter travels is a classic example of a continuous random variable. Why? Because the distance can include fractions of a mile, like 5.5 miles or 23.75 miles. We consider all possibilities, including decimals that lie in between whole numbers.
The characteristic feature of continuous variables is the smooth scale of measurement, which means there are no gaps or interruptions between values. For instance:

• The temperature of a room can be 22.3 degrees or 22.31 degrees.
• The height of a person can measure 5.7 feet or 5.75 feet.

Thus, if a variable can assume every possible value within a certain interval, it's continuous.

Range of Values

The range of values for a random variable details the set of all possible outcomes a variable can assume. In our example, the distance a commuter travels is from 0 to 150 miles.
This range includes all numbers within these endpoints, representing every possible distance a commuter might travel. We describe this mathematically using interval notation: \[X \in [0, 150]\].
The brackets "[" and "]" signify that both 0 and 150 are included. In the context of continuous variables, this range permits the variable to assume any fraction of a mile in addition to whole miles. It's essential to establish a range for operations involving probabilities, ensuring calculations are based on realistic expectations.

Finite Discrete

A finite discrete random variable is one with a specific, countable number of outcomes. Unlike continuous variables, the values here need to be distinct and separate.
For example, consider rolling a six-sided die. The outcome can be one of six definite numbers: 1, 2, 3, 4, 5, or 6. This countable set of outcomes means it's a finite discrete variable.
Key features of finite discrete variables include:

• Each outcome is distinct.
• You can list all the possible outcomes.
• Commonly measured in whole numbers.

If you can enumerate and count each distinct outcome, the random variable is likely finite discrete.

Infinite Discrete

An infinite discrete random variable differs because it has an unlimited number of outcomes, yet these outcomes are still countable. This means while you can count the possibilities, there's no end to the list.
Think of an example like the number of coins tossed until a tails appears. Theoretically, you might toss the coin forever without ever rolling tails. Here, although the number of outcomes (1, 2, 3, ...) is infinite, each outcome is a separate countable number.
Characteristics include:

• The set of outcomes shares a predictable pattern.
• Even though infinite, each outcome can be listed in a sequence.

This contrasts with continuous variables, where infinite outcomes aren't individually countable.