Problem 10 - 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 hours a child watches television on a given day

Step-by-Step Solution

Answer

The range of values for the random variable \(X\), representing the number of hours a child watches television on a given day, is \(0 \le X \le 24\). The random variable \(X\) is classified as a continuous random variable, as it can take on any value within this range.

Step 1: Determine the minimum and maximum values for X.

A child might not watch TV at all on a given day, so the minimum value for \(X\) is 0. Considering that there are 24 hours in a day, the maximum value for \(X\) would be 24 hours.

Step 2: Identify the range of values for X.

Since we have found the minimum and maximum values for \(X\), we can now represent the range of values that \(X\) may assume. The range of values for the random variable \(X\) can be denoted as \(0 \le X \le 24\).

Step 3: Classify the random variable X.

In this problem, the variable \(X\) is measuring time in hours, which can be represented as decimals or fractions of hours. Therefore, it is appropriate to classify \(X\) as a continuous random variable. To summarize, the random variable \(X\), which represents the number of hours a child watches television on a given day, can assume values within the range \(0 \le X \le 24\), and it is classified as a continuous random variable.

Key Concepts

continuous variables

Continuous variables are a key concept in statistics and probability. They are variables that can take an infinite number of values within a specified range. In the context of the exercise, the random variable \(X\) represents the number of hours a child watches television on a given day. This variable is continuous because time can be broken down into smaller and smaller fractions, like half-hours or even minutes. There is no smallest unit of time, which makes it continuous.

For example, a child could watch television for 1.5 hours or 3.2 hours; such fractional hours are all possible values of \(X\). This contrasts with discrete variables, which can only take distinct, separate values.

It's important to note that classifying a variable as continuous allows us to use different statistical tools and techniques, such as calculus-based methods, for analysis. Recognizing when data is continuous is crucial for choosing the right approach for data analysis.

range of values

The range of values for a random variable defines the spectrum of possible values that the variable can assume. In the aforementioned exercise, the random variable \(X\), which represents the number of hours a child watches television, can vary from 0 to 24 hours. This range encapsulates all possible amounts of television watched on a given day.

When describing a continuous variable like \(X\), we use inequalities: \(0 \le X \le 24\). This indicates that any value within this interval is a possible outcome for \(X\).

The concept of the range of values is fundamental in probability because it tells us where the "action" happens—where we expect to find our data. It helps in defining the probability distribution of the variable and in understanding how the values spread.

discrete vs continuous

When we think about random variables, we often categorize them as either discrete or continuous. This distinction is crucial as it affects how we analyze and interpret data.

**Discrete Variables:** These are countable and can take on only specific values. Examples include the number of students in a class or the number of leaves on a plant. They often deal with whole numbers.

**Continuous Variables:** In contrast, continuous variables can assume any value within a given range. This includes fractions and decimals, like temperature, height, or, as in our exercise, time. Since time can be measured as 1.5 hours or 22.75 hours, it is a continuous variable.

The choice between considering a random variable as discrete or continuous greatly influences the statistical methods we use. Continuous variables typically require calculus-based methods, while discrete variables can be managed with basic arithmetic. Understanding this categorization helps in proper data analysis and yields accurate results.