Problem 11 - 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 number of times an accountant takes the CPA examination before passing

Step-by-Step Solution

Answer

The range of values that the random variable \(X\) may assume is \(X = \{1, 2, 3, ... \}\), which includes all positive integers. Since the range is composed of whole numbers and infinite in length, the random variable \(X\) is classified as an infinite discrete random variable.

Step 1: Determine the minimum number of attempts

The minimum number of attempts the accountant would need to take the CPA examination before passing is 1, since they could pass it on their first try.

Step 2: Determine if there's a maximum number of attempts

There is no maximum number of attempts for taking the CPA examination before passing, as the accountant can theoretically keep taking the exam until they pass.

Step 3: Find the range of values for \(X\)

Based on Steps 1 and 2, the range of values for \(X\) includes all whole number attempts starting from 1, and it has no upper bound. In other words, the range is \(X = \{1, 2, 3, ... \}\), which represents all positive integers.

Step 4: Classify the random variable

Since the random variable \(X\) represents the number of times an accountant takes the CPA examination before passing, it can take any whole number value within the range we found in Step 3. The range is composed of whole numbers (discrete) and infinite in length as it has no upper bound. Thus, the random variable \(X\) is classified as an infinite discrete random variable.

Key Concepts

Discrete Random Variable

A discrete random variable is one that can take on a finite or countably infinite set of values. In simpler terms, it's a variable that can list its possible outcomes separately, much like counting whole numbers. Discrete random variables are often associated with counted data rather than measured data.
For example, when rolling a die, the outcome can be 1, 2, 3, 4, 5, or 6. These values are distinct and finite, making the roll a discrete event.

• Discrete variables have gaps between values. You can't have a 3.5 when rolling a die, for example.
• The outcomes are countable, like the number of students in a classroom.
• Values do not form a continuous range, so you list them individually.

In the context of the exercise, the discrete random variable is the number of CPA exam attempts, which are whole numbers like 1, 2, or 3. This setup means we can count and list each possible outcome without including every number in between.

Range of Values

The range of values for a random variable includes all the possible outcomes the variable can assume. For discrete random variables, this means listing out each potential value. Understanding the range helps us identify the nature of the variable and anticipate possible scenarios.
In the CPA exam scenario, the range is quite straightforward. The first attempt might be successful, so the lowest value is 1. The accountant can keep trying indefinitely until they pass, showing there is no upper limit. This creates a sequence of numbers starting from 1:

• Minimum value: 1 (first attempt pass).
• Possible values: 1, 2, 3, ..., n (indefinite number of attempts).

The range thus comprises all positive integers. Such clarity is crucial for classifying and interpreting the random variable properly.

Infinite Discrete

An infinite discrete random variable is a specific type of discrete variable. Unlike finite discrete variables that have a limited number of values, infinite discrete variables stretch indefinitely, listing values that can go on forever. Even though it's potentially infinite, you can still count these values.
This is evident in scenarios like the one described in the exercise, where an accountant may take the CPA exam an unbounded number of times:

• Infinite: The sequence continues with no terminal value, showing endless possible outcomes.
• Discrete: Despite infinity, you can enumerate values as 1, 2, 3, and so on.

In mathematical terms, this would be represented by the set of positive integers. Understanding this helps in statistical analysis and determining probability distributions over an unbounded countable sequence.